Problem 6
Question
Use the indicated formula from the table of integrals in this section to find the indefinite integral. $$ \int x^{2} \sqrt{x^{2}+9} d x, \text { Formula } 22 $$
Step-by-Step Solution
Verified Answer
\[ \int x^{2} \sqrt{x^{2}+9} dx = \frac{1}{5}(9+x^{4})^{3/2} + C \]
1Step 1: Identify the Integral Formula
Given the integral, it's seen that it matches formula 22: \[ \int x^n(a^2+x^{2n})^m dx = \frac{1}{2n+1}(a^2+x^{2n})^{m+1} + C\] where \(n = 2\), \(m = 1/2\), and \(a = 3\)
2Step 2: Matching and Substitution
Using the identified variables in the formula, substitute them into the formula. This will give \[\frac{1}{2*2+1}(3^2+x^{2*2})^{1/2+1} + C\]
3Step 3: Simplifying the Expression
\[= \frac{1}{5}(9+x^{4})^{3/2} + C\] This is the indefinite integral.
Other exercises in this chapter
Problem 5
Integration by parts to find the indefinite integral. $$ \int x e^{3 x} d x $$
View solution Problem 6
Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{3}^{4} \frac{1}{\sqrt{x-3}}
View solution Problem 6
Write the partial fraction decomposition for the expression. $$ \frac{7 x+5}{6\left(2 x^{2}+3 x+1\right)} $$
View solution Problem 6
Integration by parts to find the indefinite integral. $$ \int x e^{-x} d x $$
View solution