Question

find the indefinite integral and check the result by differentiating.

$$ int ( r ^ { 4 } + 2 ) ^ { 5 } r ^ { 3 } d r $$

determine an appropriate substitution to simplify the integrand. choose the correct answer below.

a. $$ ( r ^ { 4 } + 2 ) ^ { 5 } $$
b. $$ r ^ { 3 } $$
c. $$ r ^ { 4 } + 2 $$
d. $$ r ^ { 3 } ( r ^ { 4 } + 2 ) ^ { 5 } $$

Explanation:

Step1: Substitution principle

Let \(u = r^{4}+2\). Then \(du=4r^{3}dr\), and \(r^{3}dr=\frac{1}{4}du\).

Step2: Integral transformation

The integral \(\int(r^{4} + 2)^{5}r^{3}dr\) becomes \(\frac{1}{4}\int u^{5}du\).

Step3: Power - rule integration

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n
eq - 1)\), we have \(\frac{1}{4}\times\frac{u^{6}}{6}+C=\frac{u^{6}}{24}+C\).

Step4: Back - substitution

Substituting \(u = r^{4}+2\) back, we get \(\frac{(r^{4}+2)^{6}}{24}+C\).

Step5: Differentiation check

Differentiating \(y=\frac{(r^{4}+2)^{6}}{24}+C\) using the chain rule \(y^\prime=\frac{6(r^{4}+2)^{5}\times4r^{3}}{24}=(r^{4}+2)^{5}r^{3}\).

Answer:

C. \(r^{4}+2\)