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Q. 54

Expert-verified
Found in: Page 362

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# 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: $\mathrm{tan}\left(x\right)=\frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}$). $\int \mathrm{tan}xdx.$

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

See the step by step solution

## Step 1. Given Information.

Given is a integral: $\int \mathrm{tan}xdx.$

## Step 2. Formula involved.

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

## Step 3. Solving the integral.

$\int \mathrm{tan}xdx\phantom{\rule{0ex}{0ex}}=\int \frac{\mathrm{sin}x}{\mathrm{cos}x}dx\phantom{\rule{0ex}{0ex}}Lett=\mathrm{cos}x\phantom{\rule{0ex}{0ex}}dt=-\mathrm{sin}xdx\phantom{\rule{0ex}{0ex}}Puttingthevalueintheintegral\phantom{\rule{0ex}{0ex}}=\int -\frac{1}{t}dt\phantom{\rule{0ex}{0ex}}=-\mathrm{ln}t+c\phantom{\rule{0ex}{0ex}}=-\mathrm{ln}\left(\mathrm{cos}x\right)+c.$