Cosine Integral Formula

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

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

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

Other Integral Formulas of Cosine Function:

The other formulas of cosine integral with angle of sine is in the form of 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