Q. 54 - Calculus

Question

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: $\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$).

$\int \tan xdx.$

Step-by-Step Solution

Answer

The value of the given integral is $-\ln \left(\cos x\right) +c.$

Step 1. Given Information.

Given is a integral: $\int \tan xdx.$

Step 2. Formula involved.

$\int \frac{1}{t}dt = \ln t +c.$

Step 3. Solving the integral.

$$\begin{gathered} \int \tan xdx \\ = \int \frac{\sin x}{\cos x}dx \\ Let t = \cos x \\ dt = -\sin xdx \\ Putting the value in the integral \\ =\int -\frac{1}{t}dt \\ = -\ln t+c \\ = -\ln \left(\cos x\right) +c. \end{gathered}$$