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Chapter 5: Q 11. (page 495)

Fill in the blanks to complete each of the following integration formulas.
$鈭� \tan x d x =______$

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information.

Consider the given integration.

$鈭� \tan x d x$

02

Step 2. Find the integration.

Simplify the expression and find the integration.

$鈭� \tan x d x = 鈭� \frac{\sin x}{\cos x} d x$

Put $\cos x = t$ and $d t = - \sin x d x$ .

Substitute the above values in integration.

$\begin{array}{rcl} 鈭� \tan x d x & = & - 鈭� \frac{1}{t} d t \\ = & - \ln |t| + C \\ = & - \ln |\cos (x)| + C \end{array}$

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Most popular questions from this chapter

Solve the integral : $鈭� \frac{x}{e^{x^{2} + 1}} d x$

Show that if $x = \tan 鈦� u$ , then $d x = \sec^{2} 鈦� u d u$ , in the following two ways: (a) by using implicit differentiation, thinking of $u$ as a function of $x$ , and (b) by thinking of $x$ as a function of $u$ .

Solve the integral: $鈭� \frac{x^{2}}{e^{x}} d x$

Solve each of the integrals in Exercises 39鈥�74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

$鈭� \frac{x^{2} + 9}{x^{\frac{3}{2}}} d x$

Solve given definite integral.

${鈭�}_{0}^{4} 鈥� x^{3} \sqrt{x^{2} + 4} d x$

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