# 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

To prove this formula, consider

Using the derivative formula $\frac{d}{{dx}}\cos x = - \sin x$, we have

Integrating both sides of equation (i) with respect to $x$, we have

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.

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

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

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

We have integral

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