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Chapter 5: Q 37. (page 429)

Solve the integral: $\int 3 x e^{x^{2}} d x$

Short Answer

Expert verified

The required answer is $\frac{3}{2} e^{x^{2}} + c$ .

Step by step solution

Step 1. Given information.

We have given integral is $\int 3 x e^{x^{2}} d x$ .

Step 2. Solve the integration by parts .

We have,

$u = x^{2} d u = 2 x d x \frac{d u}{2} = x d x$

The formula of integration by parts is $\int u d v = u v - \int v d u$

$\int 3 x e^{x^{2}} d x = 3 \int x e^{x^{2}} d x = 3 [\frac{1}{2} \int e^{x^{2}} d x] = \frac{3}{2} e^{x^{2}} + c$

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Most popular questions from this chapter

Complete the square for each quadratic in Exercises 28–33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

$x^{2} - 4 x - 8$

True/False: Determinewhethereachofthestatementsthat follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: $f (x) = \frac{x + 1}{x - 1}$ is a proper rational function.

(b) True or False: Every improper rational function can be expressed as the sum of a polynomial and a proper rational function.

(c) True or False: After polynomial long division of p(x) by q(x), the remainder r(x) has a degree strictly less than the degree of q(x).

(d) True or False: Polynomial long division can be used to divide two polynomials of the same degree.

(e) True or False: If a rational function is improper, then polynomial long division must be applied before using the method of partial fractions.

(f) True or False: The partial-fraction decomposition of $\frac{x^{2} + 1}{x^{2} (x - 3)}$ is of the form $\frac{A}{x^{2}} + \frac{B}{x - 3}$

(g) True or False: The partial-fraction decomposition of $\frac{x^{2} + 1}{x^{2} (x - 3)}$ is of the form $\frac{B x + C}{x^{2}} + \frac{A}{x - 3}$ .

(h) True or False: Every quadratic function can be written in the form $A {(x - k)}^{2} + C$

Consider the integral $\int \frac{1}{x^{2} \sqrt{1 - x^{2}}} d x$ from the reading at the beginning of the section.

(a) Use the inverse trigonometric substitution $u = \sin^{- 1} x$ to solve this integral.

(b) Use the trigonometric substitution $x = \sin u$ to solve the integral.

Find three integrals in Exercises 27–70 for which a good strategy is to apply integration by parts twice.

Consider the integral $\int \sin x \cos x d x$ .

(a) Solve this integral by using u-substitution with $u = \sin x$ and $d u = \cos x d x$ .

(b) Solve the integral another way, using u-substitution with $u = \cos x$ and $d u = - \sin x d x$ .

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Solve the integral:∫3xex2dx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integration by parts .

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Most popular questions from this chapter

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Solve the integral: $\int 3 x e^{x^{2}} d x$