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The antiderivatives of trigonometric functions

The \( sin(x) \) function is defined as follows:

Its general antiderivative is:

The \( cos(x) \) function is defined as follows:

Its general antiderivative is:

The \( tan(x) \) function is defined as follows:

Its general antiderivative is:

The \( arcsin(x) \) is the reciprocal function of the \( sin(x) \) function, it is defined as follows:

Its general antiderivative is:

The \( arccos(x) \) function is the reciprocal function of the \( cos(x) \) function, it is defined as follows:

Its general antiderivative is:

The \( arctan(x) \) function is the reciprocal function of the \( tan(x) \) function, it is defined as follows:

Its general antiderivative is:

The \( cosec(x) \) function is defined as follows:

Its general antiderivatives are:

1. By the secant trigonometric functions

$$ \forall k \in \mathbb{Z}, \ \forall x \in \Bigl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \bigl \{ k\pi \bigr \} \Bigr],$$
$$\int^x cosec(t) \ dt = ln \left|cosec(x) -cotan(x) \right|$$

2. By applying Bioche's rules

$$ \forall k \in \mathbb{Z}, \ \forall x \in \Bigl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \bigl \{ k\pi \bigr \} \Bigr],$$
$$\int^x cosec(t) \ dt = ln \left| tan \left( \frac{x}{2}\right) \right| $$

The \( sec(x) \) function is defined as follows:

Its general antiderivatives are:

1. By the secant trigonometric functions

$$ \forall k \in \mathbb{Z}, \enspace \forall x \in \biggl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \Bigl \{ \frac{\pi}{2} + k\pi \Bigr \} \biggr], $$
$$\int^x sec(t) \ dt = ln \left|sec(x) + tan(x) \right|$$

2. By applying Bioche's rules

$$ \forall k \in \mathbb{Z}, \enspace \forall x \in \biggl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \Bigl \{ \frac{\pi}{2} + k\pi \Bigr \} \biggr], $$
$$\int^x sec(t) \ dt = ln \left| tan\left( \frac{x}{2} + \frac{\pi}{4} \right) \right| $$

The \( cotan(x) \) function is defined as follows:

Its general antiderivative is:

The \( arccosec(x) \) is the reciprocal function of the \( cosec(x) \) function, it is defined as follows:

Its general antiderivative is:

The \( arcsec(x) \) is the reciprocal function of the \( sec(x) \) function, it is defined as follows:

Its general antiderivative is:

The \( arccotan(x) \) is the reciprocal function of the \( cotan(x) \) function, it is defined as follows:

Its general antiderivative is:

The \( sinh(x) \) function is defined as follows:

Its general antiderivative is:

The \( cosh(x) \) function is defined as follows:

Its general antiderivative is:

The \( tanh(x) \) function is defined as follows:

Its general antiderivative is:

The \( arcsinh(x) \) is the reciprocal function of the \( sinh(x) \) function, it is defined as follows:

In addition, we can define it more explicitly by :

(\(\Longrightarrow\) see demonstration of it)

Its general antiderivative is:

The \( arccosh(x) \) is the reciprocal function of the \( cosh(x) \) function, it is defined as follows:

In addition, we can define it more explicitly by :

(\(\Longrightarrow\) see demonstration of it)

Its general antiderivative is:

The \( arcsinh(x) \) is the reciprocal function of the \( sinh(x) \) function, it is defined as follows:

In addition, we can define it more explicitly by :

(\(\Longrightarrow\) see demonstration of it)

Its general antiderivative is:

The \( cosech(x) \) function is defined as follows:

Its general antiderivatives are:

1. By the secant trigonometric functions

$$ \forall x \in \hspace{0.04em} \mathbb{R}^*, $$
$$\int^x cosech(t) \ dt = ln \left|cosech(x) -cotanh(x) \right|$$

2. By using the change of variable \(u = e^t\)

$$ \forall x \in \hspace{0.04em} \mathbb{R}^*,$$
$$\int^x cosech(t) \ dt = ln \left| cotanh\left(\frac{x}{2} \right) \right|$$

The \( sech(x) \) function is defined as follows:

Its general antiderivatives are:

1. By the secant trigonometric functions

$$ \forall x \in \mathbb{R}, $$
$$\int^x sech(t) \ dt = arctan(sinh(x)) $$

2. By using the change of variable \(u = e^t\)

$$ \forall x \in \mathbb{R}, $$
$$ \int^x sech(t) \ dt = 2 \ arctan(e^x) $$

The \( cotanh(x) \) function is defined as follows:

Its general antiderivative is:

The \( arccosech(x) \) is the reciprocal function of the \( cosech(x) \) function, it is defined as follows:

Its general antiderivative is:

The \( arcsech(x) \) is the reciprocal function of the \( sech(x) \) function, it is defined as follows:

Its general antiderivative is:

The \( arccotanh(x) \) is the reciprocal function of the \( cotanh(x) \) function, it is defined as follows:

Its general antiderivative is:

Demonstrations

The basic trigonometric functions\(: sin(x), cos(x), tan(x)\)

The sines function\(: {\displaystyle \int^x} sin(t) \ dt \)

The \( sin(x) \) function is defined as follows:

So by simply taking the antiderivative from each side,

The cosines function\(: cos(x)\)

The \( cos(x) \) function is defined as follows:

As well as above with the \(sin(x)\) function, we directly obtain:

The tangent function\(: {\displaystyle \int^x} tan(t) \ dt \)

The \( tan(x) \) function is defined as follows:

From this definition, we do have:

Le us set down a new variable: \(u = cos(t)\).

So,

The basic trigonometric reciprocal functions\(: arcsin(x), arccos(x), arctan(x)\)

The arcsines function\(: arcsin(x)\)

The \( arcsin(x) \) is the reciprocal function of the \( sin(x) \) function, it is defined as follows:

From this definition, let us perfom an integration by parts with :

We do have:

And as a result,

The arccosines function\(: arccos(x)\)

The \( arccos(x) \) function is the reciprocal function of the \( cos(x) \) function, it is defined as follows:

As well as above, we perform an integration by parts with: :

We do have:

As a result we do have,

The arctangent function\(: {\displaystyle \int^x} arctan(t) \ dt \)

The \( arctan(x) \) function is the reciprocal function of the \( tan(x) \) function, it is defined as follows:

From this definition, let us perfom an integration by parts with :

We do have:

And as a result,

The secant trigonometric functions\( : cosec(x), sec(x), cotan(x)\)

The cosecant function\(: {\displaystyle \int^x} cosec(t) \ dt \)

1. By the secant trigonometric functions

The \( cosec(x) \) function is defined as follows:

$$ \forall k \in \mathbb{Z}, \ \forall x \in \Bigl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \bigl \{ k\pi \bigr \} \Bigr], \enspace f(x) = cosec(x) = \frac{1}{sin(x)} $$

We firstly notice that:

$$cosec(x) = cosec(x)\frac{cosec(x) - cotan(x)}{cosec(x) - cotan(x)}$$
$$cosec(x) = \frac{cosec^2(x) - cosec(x)cotan(x)}{cosec(x) - cotan(x)}$$

But,

$$ \Biggl \{ \begin{gather*} cosec^2(x) = -cotan(x)' \\ -cosec(x)cotan(x) = cosec(x)' \end{gather*} $$

Now we have,

$$cosec(x) = \frac{-cotan'(x) + cosec'(x) }{cosec(x) -cotan(x)}$$
$$cosec(x) = \frac{(cosec(x) -cotan(x))'}{cosec(x) -cotan(x)}$$

Then, we can easily integrate it and:

$$\int^x cosec(t) \ dt = \int^x \frac{\bigl(cosec(t) -cotan(t) \bigr)'}{cosec(t) -cotan(t)} \ dt $$

As a result,

$$ \forall k \in \mathbb{Z}, \ \forall x \in \Bigl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \bigl \{ k\pi \bigr \} \Bigr],$$
$$\int^x cosec(t) \ dt = ln \left|cosec(x) -cotan(x) \right|$$

2. By applying Bioche's rules

By applying Bioche's rules, we can put as a change of variable:

$$ u = tan\left(\frac{t}{2} \right) $$
$$ \left \{ \begin{gather*} u = tan\left(\frac{t}{2} \right) \\ du = \frac{1}{2} \left(1 + tan^2\left(\frac{t}{2} \right) \right) dt \Longleftrightarrow du = \frac{1}{2} \left(1 + u^2 \right) \ dt \Longleftrightarrow dt = \frac{2du}{1 + u^2} \end{gather*} \right \} $$

Then the the integral:

$$\int^x cosec(t) \ dt = \int^x \frac{1}{sin(t)} \ dt $$

becomes:

$$\int^x cosec(t) \ dt = \int^x \frac{1 + u^2}{2u} \ \frac{2du}{1 + u^2} $$
$$\int^x cosec(t) \ dt = \int^x \frac{du}{u} $$

And finally,

$$ \forall k \in \mathbb{Z}, \ \forall x \in \Bigl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \bigl \{ k\pi \bigr \} \Bigr],$$
$$\int^x cosec(t) \ dt = ln \left| tan \left( \frac{x}{2}\right) \right| $$

The secant function\(: {\displaystyle \int^x} sec(t) \ dt \)

1. By the secant trigonometric functions

The \( sec(x) \) function is defined as follows:

$$ \forall k \in \mathbb{Z}, \enspace \forall x \in \biggl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \Bigl \{ \frac{\pi}{2} + k\pi \Bigr \} \biggr], \enspace f(x) = sec(x) = \frac{1}{cos(x)} $$

First of all, we notice that:

$$sec(x) = sec(x)\frac{sec(x) + tan(x)}{sec(x) + tan(x)}$$
$$sec(x) = \frac{sec^2(x) + sec(x)tan(x)}{sec(x) + tan(x)}$$

But,

$$ \Biggl \{ \begin{gather*} sec^2(x) = tan'(x) \\ sec(x)tan(x)= sec'(x) \end{gather*} $$

Therefore,

$$sec(x) = \frac{tan'(x) + sec'(x) }{sec(x) + tan(x)}$$
$$sec(x) = \frac{(sec(x) + tan(x))'}{sec(x) + tan(x)}$$

Now we can easily integrate it and:

$$\int^x sec(t) \ dt = \int^x \frac{(sec(x) + tan(x))'}{sec(x) + tan(x)} \ dt $$

And a result we do obtain,

$$ \forall k \in \mathbb{Z}, \enspace \forall x \in \biggl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \Bigl \{ \frac{\pi}{2} + k\pi \Bigr \} \biggr], $$
$$\int^x sec(t) \ dt = ln \left|sec(x) + tan(x) \right|$$

2. By applying Bioche's rules

In the same way as above, we put the same change of variable:

$$ u = tan\left(\frac{t}{2} \right) $$
$$ \left \{ \begin{gather*} u = tan\left(\frac{t}{2} \right) \\ du = \frac{1}{2} \left(1 + tan^2\left(\frac{t}{2} \right) \right) dt \Longleftrightarrow du = \frac{1}{2} \left(1 + u^2 \right) \ dt \Longleftrightarrow dt = \frac{2du}{1 + u^2} \end{gather*} \right \} $$

Then the the integral:

$$\int^x sec(t) \ dt = \int^x \frac{1}{cos(t)} \ dt $$

becomes:

$$\int^x sec(t) \ dt = \int^x \frac{1 + u^2}{1 - u^2} \ \frac{2du}{1 + u^2} $$
$$\int^x sec(t) \ dt = 2\int^x \frac{du}{1 - u^2} $$
$$\int^x sec(t) \ dt = 2\int^x \frac{du}{(1 - u)(1 + u)} $$

After having broken it in simple elements, we do have:

$$\int^x sec(t) \ dt = 2 \times \frac{1}{2} ln \left| \frac{1 + u}{1 - u} \right| $$

Now, by rehabilitating the starting variable:

$$\int^x sec(t) \ dt = ln \left| \frac{1 + tan\left(\frac{x}{2} \right)}{1 - tan\left(\frac{x}{2} \right)} \right| $$

But we know from the addition trigonometric formulas, that:

$$ tan(\alpha + \beta) = \frac{tan(\alpha) + tan(\beta)}{1 - tan(\alpha)tan(\beta) }$$
$$\int^x sec(t) \ dt = ln \left| \frac{ tan\left(\frac{\pi}{4} \right) + tan\left(\frac{x}{2} \right)}{1 - tan\left(\frac{\pi}{4} \right) tan\left(\frac{x}{2} \right)} \right| $$

We finally obtain that,

$$ \forall k \in \mathbb{Z}, \enspace \forall x \in \biggl[ \mathbb{R} \hspace{0.2em} \backslash \hspace{0.2em} \Bigl \{ \frac{\pi}{2} + k\pi \Bigr \} \biggr], $$
$$\int^x sec(t) \ dt = ln \left| tan\left( \frac{x}{2} + \frac{\pi}{4} \right) \right| $$

The cotangent function\(: {\displaystyle \int^x} cotan(t) \ dt \)

The \( cotan(x) \) function is defined as follows:

From this definition:

So,

Let us set down: \(u = sin(t)\).

As a result we do have,

The secant trigonometric reciprocal functions\(: arccosec(x), arcsec(x), arccotan(x)\)

The arccosecant function\(: {\displaystyle \int^x} arccosec(t) \ dt \)

The \( arccosec(x) \) is the reciprocal function of the \( cosec(x) \) function, it is defined as follows:

From this definition, let us perfom an integration by parts with :

We do have:

We already calculated this integral above:

And finally,

The arcsecant function\(: {\displaystyle \int^x} arcsec(t) \ dt \)

The \( arcsec(x) \) is the reciprocal function of the \( sec(x) \) function, it is defined as follows:

From this definition, by performing the same integration by parts as the \(arccosec(x)\) function above:

We directly obtain,

The arccotangent function\(: {\displaystyle \int^x} arccotan(t) \ dt \)

The \( arccotan(x) \) is the reciprocal function of the \( cotan(x) \) function, it is defined as follows:

From this definition, by performing the same integration by parts as the \(arccosec(x)\) function above:

We directly obtain,

The hyperbolic function\(: sinh(x), cosh(x), tanh(x)\)

The hyperbolic sines function\(: {\displaystyle \int^x} sinh(t) \ dt \)

The \( sinh(x) \) function is defined as follows:

As well as above with the \(sin(x)\) function, we directly obtain:

The hyperbolic cosines function\(: {\displaystyle \int^x} cosh(t) \ dt \)

The \( cosh(x) \) function is defined as follows:

As well as above with the \(sinh(x)\) function, we directly obtain:

The hyperbolic tangent function\(: {\displaystyle \int^x} tanh(t) \ dt \)

The \( tanh(x) \) function is defined as follows:

From this definition, we do have:

As well as above with the \(tan(x)\) function, we set down: \(u = cosh(t)\).

And we easily obtain,

The hyperbolic reciprocal functions\(: arcsinh(x), arccosh(x) ,arctanh(x)\)

The hyperbolic arcsines function\(: {\displaystyle \int^x} arcsinh(t) \ dt \)

The \( arcsinh(x) \) is the reciprocal function of the \( sinh(x) \) function, it is defined as follows:

In addition, we can define it more explicitly by :

(\(\Longrightarrow\) see demonstration of it)

As well as above, we perform an integration by parts with:

We do have:

And as a result,

The hyperbolic arccosines function\(: {\displaystyle \int^x} arccosh(t) \ dt \)

The \( arccosh(x) \) is the reciprocal function of the \( cosh(x) \) function, it is defined as follows:

In addition, we can define it more explicitly by :

(\(\Longrightarrow\) see demonstration of it)

As well as above, we perform an integration by parts with:

We do have:

And finally,

The hyperbolic arctangent function\(: {\displaystyle \int^x} arctanh(t) \ dt \)

The \( arctanh(x) \) is the reciprocal function of the \( tanh(x) \) function, it is defined as follows:

In addition, we can define it more explicitly by :

(\(\Longrightarrow\) see demonstration of it)

From this definition, let us perfom an integration by parts with :

We do have:

And finally,

The hyperbolic secant functions\(: cosech(x), sech(x), cotanh(x)\)

The hyperbolic cosecant function\(: {\displaystyle \int^x} cosech(t) \ dt \)

The \( cosech(x) \) function is defined as follows:

1. By the secant trigonometric functions

By applying the same reasoning as above with the \(cosec(x) \) function :

$$ \Biggl \{ \begin{gather*} cosech^2(x) = -cotanh(x)' \\ -cosech(x)cotanh(x) = cosech(x)' \end{gather*} $$

We directly obtain that:

$$ \forall x \in \hspace{0.04em} \mathbb{R}^*, $$
$$\int^x cosech(t) \ dt = ln \left|cosech(x) -cotanh(x) \right|$$

2. By using the change of variable \(u = e^t\)

We now use this change of variable:

$$ \left \{ \begin{gather*} u = e^t \\ du = e^t \ dt \Longleftrightarrow dt = \frac{du}{u} \end{gather*} \right \} $$

Then the the integral:

$$\int^x cosech(t) \ dt = \int^x \frac{1}{sinh(t)} \ dt = \int^x \frac{2}{e^t - e^{-t}} \ dt $$

becomes:

$$\int^x cosech(t) \ dt = \int^x \frac{2}{u - u^{-1}} \ \frac{du}{u} $$
$$\int^x cosech(t) \ dt = 2 \int^x \frac{du}{u^2 - 1} $$

After having broken it in simple elements, we do have:

$$\int^x cosech(t) \ dt = 2 \times \frac{1}{2} ln \left| \frac{u + 1}{u - 1} \right| $$

Now, by rehabilitating the starting variable:

$$\int^x cosech(t) \ dt = 2 \times \frac{1}{2} ln \left| \frac{e^x + 1}{e^x - 1} \right| $$
$$\int^x cosech(t) \ dt = 2 \times \frac{1}{2} ln \left| \frac{(e^{\frac{x}{2}} + e^{-\frac{x}{2}})}{e^{\frac{x}{2}}(e^{\frac{x}{2}} - e^{-\frac{x}{2}})} \right| $$
$$\int^x cosech(t) \ dt = 2 \times \frac{1}{2} ln \left| \frac{e^{\frac{x}{2}} + e^{-\frac{x}{2}}}{e^{\frac{x}{2}} - e^{-\frac{x}{2}}} \right| $$
$$\int^x cosech(t) \ dt = ln \left| \left( \frac{1}{tanh \left(\frac{x}{2} \right) } \right) \right| $$

We finally obtain that :

$$ \forall x \in \hspace{0.04em} \mathbb{R}^*,$$
$$\int^x cosech(t) \ dt = ln \left| cotanh\left(\frac{x}{2} \right) \right|$$

The hyperbolic secant function\(: {\displaystyle \int^x} sech(t) \ dt \)

The \( sech(x) \) function is defined as follows:

1. By the secant trigonometric functions

$$ \int^x sech(t) \ dt = \int^x \frac{1}{cosh(t)} \ dt $$
$$ \int^x sech(t) \ dt = \int^x \frac{cosh(t)}{cosh^2(t)} \ dt $$
$$ \int^x sech(t) \ dt = \int^x \frac{cosh(t)}{1 + sinh^2(t)} \ dt $$

Let us set down the new variable: \(u = sinh(t)\).

Now we have:

$$ \begin{gather*} \int^x sech(t) \ dt = \int^x \frac{du}{1 + u^2} \end{gather*} $$
$$ with \enspace \Biggl \{ \begin{gather*} u = sinh(t) \\ du = cosh(t) \ dt \end{gather*} $$

And finally,

$$ \forall x \in \mathbb{R}, $$
$$\int^x sech(t) \ dt = arctan(sinh(x)) $$

2. By using the change of variable \(u = e^t\)

In the same manner as previously, we put the following change of variable:

$$ \left \{ \begin{gather*} u = e^t \\ du = e^t \ dt \Longleftrightarrow dt = \frac{du}{u} \end{gather*} \right \} $$

Then the the integral:

$$\int^x sech(t) \ dt = \int^x \frac{1}{cosh(t)} \ dt = \int^x \frac{2}{e^t + e^{-t}} \ dt $$

becomes:

$$\int^x sech(t) \ dt = \int^x \frac{2}{u + u^{-1}} \ \frac{du}{u} $$
$$\int^x sech(t) \ dt = 2 \int^x \frac{du}{u^2 + 1} $$
$$\int^x sech(t) \ dt = 2 \ arctan(u) $$

Now, by rehabilitating the starting variable, we finally obtain that:

$$ \forall x \in \mathbb{R}, $$
$$ \int^x sech(t) \ dt = 2 \ arctan(e^x) $$

The hyperbolic cotangent function\(: {\displaystyle \int^x} cotanh(t) \ dt \)

The \( cotanh(x) \) function is defined as follows:

From this definition:

As well as above, we set a new variable: \(u = sinh(t)\).

We finally obtain,

The hyperbolic secant reciprocal functions\(: arccosech(x), arcsech(x),arccotanh(x)\)

The hyperbolic arccosecant function\(: {\displaystyle \int^x} arccosech(t) \ dt \)

The \( arccosech(x) \) is the reciprocal function of the \( cosech(x) \) function, it is defined as follows:

From this definition, let us perfom an integration by parts with :

We already calculated this integral above:

And finally,

The hyperbolic arcsecant function\(: {\displaystyle \int^x} arcsech(t) \ dt \)

The \( arcsech(x) \) is the reciprocal function of the \( sech(x) \) function, it is defined as follows:

From this definition, by performing the same integration by parts as the \(arccosech(x)\) function above:

We directly obtain,

The hyperbolic arccotangent function\(: {\displaystyle \int^x} arccotanh(t) \ dt \)

The \( arccotanh(x) \) is the reciprocal function of the \( cotanh(x) \) function, it is defined as follows:

From this definition, by performing the same integration by parts as the \(arccosech(x)\) function above:

We directly obtain,

Antiderivatives of trigonometric functions recap table

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