Tangent Integral Formula

In this tutorial we will prove the formula of tangent integral which is also an important formula in integral calculus; this integral belongs to trigonometric formulae.

The integration of tangent function is of the form

\int  {\tan xdx = } \ln \sec x + c


\int {\tan xdx =   - } \ln \cos x + c

To prove this formula, consider

\frac{d}{{dx}}\left[  {\ln \sec x + c} \right] = \frac{d}{{dx}}\ln \sec x + \frac{d}{{dx}}c

Using the derivative formulas \frac{d}{{dx}}\ln x = \frac{1}{x} and \frac{d}{{dx}}\sec x = \sec \tan x, we have

\begin{gathered} \frac{d}{{dx}}\left[ {\ln \sec x + c} \right]  = \frac{1}{{\sec x}}\frac{d}{{dx}}\sec x + 0 \\ \Rightarrow \frac{d}{{dx}}\left[ {\ln \sec x  + c} \right] = \frac{1}{{\sec x}}\left( {\sec x\tan x} \right) \\ \Rightarrow \frac{d}{{dx}}\left[ {\ln \sec x  + c} \right] = \tan x \\ \Rightarrow \tan x = \frac{d}{{dx}}\left[  {\ln \sec x + c} \right] \\ \tan xdx = d\left[ {\ln \sec x + c} \right]\,\,\,\,{\text{  -  -   - }}\left( {\text{i}} \right) \\ \end{gathered}

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

\int  {\tan xdx}  = \int {d\left[ {\ln \sec 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  {\tan xdx = } \ln \sec x + c

Further we can prove this formula in another form as,

\begin{gathered} \int {\tan xdx = } \ln \sec x + c \\ \Rightarrow \int {\tan xdx = } \ln  \frac{1}{{\cos x}} + c \\ \Rightarrow \int {\tan xdx = } \ln {\left(  {\cos x} \right)^{ - 1}} + c \\ \Rightarrow \int {\tan xdx = }  - \ln \left( {\cos x} \right) + c \\ \end{gathered}

Alternate Proof:
We have given integration of the form

\int  {\tan xdx = \int {\frac{{\sin x}}{{\cos x}}dx} }

Here we have f\left( x  \right) = \cos x then f'\left( x  \right) =  - \sin x

\int  {\tan xdx =  - \int {\frac{{ - \sin  x}}{{\cos x}}dx} }

Using the formula of integration, \int {\frac{{f'\left( x \right)}}{{f\left( x  \right)}}dx = \ln f\left( x \right) + c}

\int  {\tan xdx =  - \ln \cos x + c}

Other Integral Formulas of Tangent Function:

The other formulas of tangent integral with angle of sine is in the form of function are given as

1. \int  {\tan axdx = \frac{1}{a}\ln \sec ax + c}

2. \int  {\tan axdx =  - \frac{1}{a}\ln \cos ax +  c}
3. \int  {\tan f\left( x \right)f'\left( x \right)dx = \ln \sec f\left( x \right) + c}
4. \int  {\tan f\left( x \right)f'\left( x \right)dx =   - \ln \cos f\left( x \right) + c}

Example: Evaluate the integral \int {\tan 6xdx} with respect to x

We have integral

I =  \int {\tan 6xdx}

Using the formula \int  {\tan axdx = \frac{1}{a}\ln \sec ax + c} , we have

\int  {\tan 6xdx}  = \frac{1}{6}\ln \sec 6x +  c