# Sine Integral Formula

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

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

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

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

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

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.
$\int {\sin xdx = } – \cos x + c$

Other Integral Formulae of the Sine Function

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

1. $\int {\sin axdx = – \frac{{\cos ax}}{a}} + c$

2. $\int {\sin f\left( x \right)f’\left( x \right)dx = – \cos f\left( x \right) + c}$

Example: Evaluate the integral $\int {\sin 4xdx}$ with respect to $x$

We have integral $I = \int {\sin 4xdx}$

Using the formula $\int {\sin axdx = – \frac{{\cos ax}}{a}} + c$, we have
$\int {\sin 4xdx} = – \frac{{\cos 4x}}{4} + c$

Example: Evaluate the integral $\int {\frac{{\sin \sqrt x }}{{2\sqrt x }}dx}$ with respect to $x$

We have integral $I = \int {\frac{{\sin \sqrt x }}{{2\sqrt x }}dx}$
$I = \int {\sin \sqrt x \frac{1}{{2\sqrt x }}dx}$

Using the formula $\int {\sin f\left( x \right)f’\left( x \right)dx = \sin f\left( x \right) + c}$, we have
$\int {\frac{{\sin \sqrt x }}{{2\sqrt x }}dx} = – \cos \sqrt x + c$