Question

find the indefinite integral and check the result by differentiating.
intleft(x^{2}-1
ight)^{12}(2 x) d x
determine an appropriate substitution to simplify the integrand. choose the correct
answer below.
a. ( 2 xleft(x^{2}-1
ight)^{12} )
b. ( 2left(x^{2}-1
ight)^{12} )
c. ( left(x^{2}-1
ight)^{12} )
d. ( x^{2}-1 )
find the indefinite integral.
intleft(x^{2}-1
ight)^{12}(2 x) d x=square

Explanation:

Step1: Substitution

Let \(u = x^{2}-1\), then \(du=2x dx\).
The integral \(\int(x^{2} - 1)^{12}(2x)dx\) becomes \(\int u^{12}du\).

Step2: Integrate

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), for \(\int u^{12}du\), we have \(\frac{u^{13}}{13}+C\).

Step3: Back - substitution

Substitute \(u = x^{2}-1\) back into the result. So \(\frac{(x^{2}-1)^{13}}{13}+C\).

Step4: Check by differentiating

Differentiate \(y=\frac{(x^{2}-1)^{13}}{13}+C\) using the chain rule.
Let \(t=x^{2}-1\), then \(y = \frac{t^{13}}{13}+C\). \(\frac{dy}{dt}=t^{12}\) and \(\frac{dt}{dx}=2x\).
By the chain rule \(\frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx}=(x^{2}-1)^{12}(2x)\).

Answer:

\(\frac{(x^{2}-1)^{13}}{13}+C\)