Please answer 11b please and thank you! 11) Find each integral by substitutiona) Jóx(3x +1)'dx10x+1b) I x +x+8)

(a)∫6x3x2+18dx Let 3x2+1=u The derivative of the above equation. 6x+0=dudx6xdx=du Substitute the…

Answered: 11) Find each integral by substitution…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Please answer 11b please and thank you!

11) Find each integral by substitution
a) Jóx(3x +1)'dx
10x+1
b) I x +x+8)

Expert Solution

Step 1

(a) $\int 6 x {(3 x^{2} + 1)}^{8} d x$

Let

$3 x^{2} + 1 = u$

The derivative of the above equation.

$\begin{array}{rcl} 6 x + 0 & = & \frac{d u}{d x} \\ 6 x d x & = & d u \end{array}$

Substitute the obtained value in the integrand and solve the integration.

$\begin{array}{rcl} \int 6 x {(3 x^{2} + 1)}^{8} d x & = & \int {(3 x^{2} + 1)}^{8} (6 x d x) \\ = & \int u^{8} d u \\ = & [\frac{u^{9}}{9}] + C \end{array}$

Substitute 3x²+1 for u in the above equation.

$\begin{array}{rcl} \int 6 x {(3 x^{2} + 1)}^{8} d x & = & [\frac{{(3 x^{2} + 1)}^{9}}{9}] + C \\ = & \frac{{(3 x^{2} + 1)}^{9}}{9} + C \end{array}$

Thus, the value of integration is $\begin{array}{rcl} \frac{{(3 x^{2} + 1)}^{9}}{9} \end{array} + C$ .

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