Chapter 5: Q. 55 (page 429)

Solve each of the integrals in Exercises 27–70. Some integrals require integration by parts, and some do not. (The last two exercises involve hyperbolic functions.)
$\int x^{3} \sin x^{2} d x$

Short Answer

Expert verified

The value is $- \frac{x^{2}}{2} \cos (x^{2}) + \frac{1}{2} \sin (x^{2}) + C .$

Step by step solution

Step 1. Given information.

The given integral is $\int x^{3} \sin x^{2} d x$ .

Step 2. Substitution.

Now,

$u = x^{2} d u = 2 x d x \int x^{3} \sin x^{2} d x = \int \frac{u}{2} \sin (u) d u = \frac{1}{2} \int u \sin (u) d u$

Step 3. Value of the integral.

Again,

$let w = u, d v = \sin x^{2} d x . d v = \sin^{2} x d x v = \frac{x}{2} - \frac{1}{4} \sin 2 x N o w, \frac{1}{2} \int u \sin (u) d u = \frac{1}{2} (u (- \cos u)) - \int (- \cos u) d u = \frac{1}{2} (- u \cos u) + \int \cos u d u = - \frac{u}{2} \cos u + \frac{1}{2} \int \cos u d u = - \frac{u}{2} \cos u + \frac{1}{2} \sin u = - \frac{x^{2}}{2} \cos (x^{2}) + \frac{1}{2} \sin (x^{2}) + C$

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Solve each of the integrals in Exercises 27–70. Some integrals require integration by parts, and some do not. (The last two exercises involve hyperbolic functions.)
$\int x^{3} \sin x^{2} d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Substitution.

Step 3. Value of the integral.

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Solve each of the integrals in Exercises 27–70. Some integrals require integration by parts, and some do not. (The last two exercises involve hyperbolic functions.)∫x3sinx2dx

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

Theoretical and Mathematical Physics

Statistics

Pure Maths

Decision Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Solve each of the integrals in Exercises 27–70. Some integrals require integration by parts, and some do not. (The last two exercises involve hyperbolic functions.)
$\int x^{3} \sin x^{2} d x$