# Sine Integral Formula

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

The integration of 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 derivative, so the integral sign $\int {}$and $\frac{d}{{dx}}$ on the right side will cancel each other, i.e.

Other Integral Formulas of Sine Function:
The other formulas of sine integral with angle of sine is in the form of 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