Question

Fill in the blanks to complete each of the following integration formulas.

$\int \tan xdx=\_\_\_\_\_\_$

Short answer

The obtained result is$-\ln \left|\cos \left(x\right)\right|+C$.

Step-by-step solution

Step 1. Given information.

Consider the given integration.

$\int \tan xdx$

Step 2. Find the integration.

Simplify the expression and find the integration.

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

Put $\cos x=t$and $dt=-\sin xdx$.

Substitute the above values in integration.

$\begin{array}{rcl}\int \tan xdx& =& -\int \frac{1}{t}dt\\ & =& -\ln \left|t\right|+C\\ & =& -\ln \left|\cos \left(x\right)\right|+C\end{array}$