# Cosine Integral Formula

Integration of the cosine function is an important integral formula in integral calculus. This integral belongs to the trigonometric formulae.

The integration of the cosine function is of the form
$\int {\cos xdx = } \sin x + c$

To prove this formula, consider
$\frac{d}{{dx}}\left[ {\sin x + c} \right] = \frac{d}{{dx}}\sin x + \frac{d}{{dx}}c$

Using the derivative formula $\frac{d}{{dx}}\sin x = \cos x$, we have
$\begin{gathered} \frac{d}{{dx}}\left[ {\sin x + c} \right] = \frac{d}{{dx}}\sin x + \frac{d}{{dx}}c \\ \Rightarrow \frac{d}{{dx}}\left[ {\sin x + c} \right] = \cos x + 0 \\ \Rightarrow \frac{d}{{dx}}\left[ {\sin x + c} \right] = \cos x \\ \Rightarrow \cos x = \frac{d}{{dx}}\left[ {\sin x + c} \right] \\ \Rightarrow \cos xdx = d\left[ {\sin x + c} \right]\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Integrating both sides of equation (i) with respect to $x$, we have
$\int {\cos xdx} = \int {d\left[ {\sin x + c} \right]}$

As we know that by definition integration is the inverse process of the derivative, the integral sign $\int {}$and $\frac{d}{{dx}}$ on the right side will cancel each other out, i.e.
$\int {\cos xdx = } \sin x + c$

Other Integral Formulae of the Cosine Function

The other formulae of cosine integral with an angle of sine in the form of a function are given as

1. $\int {\cos axdx = \frac{{\sin ax}}{a}} + c$
2. $\int {\cos f\left( x \right)f’\left( x \right)dx = \sin f\left( x \right) + c}$

Example: Evaluate the integral $\int {\cos 7xdx}$ with respect to $x$

We have integral $I = \int {\cos 7xdx}$

Using the formula $\int {\cos axdx = \frac{{\sin ax}}{a}} + c$, we have
$\int {\cos 7xdx} = \frac{{\sin 7x}}{7} + c$

Example: Evaluate the integral $\int {\cos \left( {\sin x} \right)\cos xdx}$ with respect to $x$

We have integral $I = \int {\cos \left( {\sin x} \right)\cos xdx}$
$I = \int {\cos \left( {\sin x} \right)\cos xdx}$

Using the formula $\int {\cos f\left( x \right)f’\left( x \right)dx = \sin f\left( x \right) + c}$, we have
$\int {\cos \left( {\sin x} \right)\cos xdx} = \sin \left( {\sin x} \right) + c$