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Chapter 5: Q 64. (page 429)

Solve the integral: $鈭� x^{3} e^{x^{2}} d x$

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information.

We have given integral is $鈭� x^{3} e^{x^{2}} d x$ .

02

Step 2. Solve the integration by parts.

We have,

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

So, we have

$鈭� x^{3} e^{x^{2}} d x = \frac{u e^{u}}{2} d x$

Now, $w = u d w = d u$

and $d v = e^{u} d u v = 鈭� e^{u} d u v = e^{u}$

The formula of integration by parts is $鈭� u d v = u v - 鈭� v d u$ .

$\frac{1}{2} 鈭� u e^{u} d u = \frac{1}{2} (u . e^{u} - 鈭� e^{u} d u) = \frac{u e^{u}}{2} - \frac{1}{2} 鈭� e^{u} d u = \frac{u e^{u}}{2} - \frac{e^{u}}{2} + c = \frac{x^{2} e^{x^{2}}}{2} - \frac{e^{x^{2}}}{2} + c$

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Most popular questions from this chapter

Solve the integral: $鈭� \ln \sqrt{x} d x$

Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial $x^{2} - x + 3$ as the quotient above the top line, and the polynomial 3x 鈭� 1 at the bottom as the remainder. Then $p (x) =___a n d \frac{p (x)}{q (x)} =____$

Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.

$鈭� \frac{x^{4} - 3}{2 + 3 x^{2}} d x$

Solve the integral: $鈭� \ln (x^{3}) d x$ .

For each integral in Exercises 5鈥�8, write down three integrals that will have that form after a substitution of variables.

$鈭� u^{2} d u$

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