Question

find the indefinite integral and check the result by differentiating.
intleft(r^{4}+2
ight)^{5} r^{3} d r
determine an appropriate substitution to simplify the integrand. choose the correct answer below.
a. ( left(r^{4}+2
ight)^{5} )
b. ( r^{3} )
c. ( r^{4}+2 )
d. ( r^{3}left(r^{4}+2
ight)^{5} )
find the indefinite integral.
intleft(r^{4}+2
ight)^{5} r^{3} d r= )

Explanation:

Step1: Use substitution

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

Step2: Substitute into the integral

\(\int(r^{4} + 2)^{5}r^{3}dr=\int u^{5}\cdot\frac{1}{4}du\)

Step3: Integrate with respect to \(u\)

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n
eq - 1)\), we have \(\frac{1}{4}\int u^{5}du=\frac{1}{4}\cdot\frac{u^{6}}{6}+C=\frac{u^{6}}{24}+C\)

Step4: Substitute back \(u = r^{4}+2\)

\(\frac{(r^{4}+2)^{6}}{24}+C\)

Step5: Check by differentiating

Differentiate \(y=\frac{(r^{4}+2)^{6}}{24}+C\) using the chain rule \((f(g(r)))^\prime=f^\prime(g(r))\cdot g^\prime(r)\). Let \(f(u)=\frac{u^{6}}{24}\), \(u = g(r)=r^{4}+2\). \(f^\prime(u)=\frac{6u^{5}}{24}=\frac{u^{5}}{4}\), \(g^\prime(r) = 4r^{3}\). Then \(y^\prime=\frac{(r^{4}+2)^{5}}{4}\cdot4r^{3}=(r^{4}+2)^{5}r^{3}\)

Answer:

\(\frac{(r^{4}+2)^{6}}{24}+C\)