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Chapter 5: Q.71 (page 465)

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.
$鈭� e^{2 x} \sqrt{e^{2 x} + 1} d x$

Short Answer

Expert verified

The value of integral is $\frac{1}{3} {(e^{2 x} + 1)}^{\frac{3}{2}} + C$

Step by step solution

01

Step 1. Given information

An expression is given as $鈭� e^{2 x} \sqrt{e^{2 x} + 1} d x$

02

Step 2.Solving integral

Let

$u = e^{2 x} + 1 d u = 2 e^{2 x} d x$

Now substitute the value and simplify as,

$鈭� e^{2 x} \sqrt{e^{2 x} + 1} d x = 鈭� \frac{\sqrt{u}}{2} d u = \frac{1}{2} 鈭� \sqrt{u} d u = \frac{(u)^{\frac{3}{2}}}{3} = \frac{1}{3} {(e^{2 x} + 1)}^{\frac{3}{2}} + C$

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Most popular questions from this chapter

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.

$鈭� \frac{1}{{(x^{2} + 4)}^{\frac{3}{2}}} d x$

Why don鈥檛 we ever have cause to use the trigonometric substitution $x = \cos u$ ?

Explain why $鈭� \frac{2 x}{x^{2} + 1} d x$ and $鈭� \frac{1}{x \ln x} d x$ are essentially the same integral after a change of variables.

Solve the integral

$鈭� \sqrt{4 - x} \sqrt{x - 2} d x$

Suppose $u (x) = x^{2}$ . Calculate and compare the values of the following definite integrals:

role="math" localid="1648786835678" ${鈭�}_{- 1}^{5} u^{2} d u, {鈭�}_{x = - 1}^{x = 5} u^{2} d u and {鈭�}_{u (- 1)}^{u (5)} u^{2} d u$

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