Q. 56 Solve each of the integrals in E... [FREE SOLUTION]

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Chapter 5: Q. 56 (page 429)

Solve each of the integrals in Exercises 27鈥�70. Some integrals require integration by parts, and some do not. (The last two exercises involve hyperbolic functions.)
$鈭� \arcsin 2 x d x$

Short Answer

Expert verified

The value of the integral is $x \arcsin (2 x) + \frac{1}{2} \sqrt{- 4 x^{2} + 1} .$

Step by step solution

Step 1. Given information.

The given integral is $鈭� \arcsin 2 x d x .$

Step 2. Substitution.

Now,

$u = 2 x; d u = 2 d x 鈭� \arcsin 2 x d x = \frac{1}{2} 鈭� \arcsin (u) d u w = \arcsin u; d w = \frac{d u}{\sqrt{- u^{2} + 1}} v = u; d v = d u$

Step 3. Value of the integral.

The value of the integral is,

$\frac{1}{2} 鈭� \arcsin (u) d u = \frac{1}{2} ((u \arcsin u) - 鈭� \frac{u}{\sqrt{- u^{2} + 1}} d u) v = - u^{2} + 1; d v = - 2 u d u \frac{1}{2} ((u \arcsin u) - 鈭� \frac{u}{\sqrt{- u^{2} + 1}} d u) = \frac{u}{2} \arcsin u - \frac{1}{2} 鈭� (- \frac{1}{2 \sqrt{v}}) d v = \frac{u}{2} \arcsin u + \frac{1}{4} 鈭� (\frac{1}{\sqrt{v}}) d v = \frac{u}{2} \arcsin u + \frac{1}{4} (\frac{v^{\frac{1}{2}}}{\frac{1}{2}}) = x \arcsin (2 x) + \frac{1}{2} \sqrt{- 4 x^{2} + 1}$

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91影视

Solve each of the integrals in Exercises 27鈥�70. Some integrals require integration by parts, and some do not. (The last two exercises involve hyperbolic functions.)
$鈭� \arcsin 2 x d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Substitution.

Step 3. Value of the integral.

One App. One Place for Learning.

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91影视

Solve each of the integrals in Exercises 27鈥�70. Some integrals require integration by parts, and some do not. (The last two exercises involve hyperbolic functions.)鈭�arcsin2xdx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Substitution.

Step 3. Value of the integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Geometry

Mechanics Maths

Applied Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Solve each of the integrals in Exercises 27鈥�70. Some integrals require integration by parts, and some do not. (The last two exercises involve hyperbolic functions.)
$鈭� \arcsin 2 x d x$