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

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

\int {\sin xdx = } - \cos x + c

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

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