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 )

Explanation:

Step1: Recall substitution rule

In substitution method for integration \(\int f(g(x))g^{\prime}(x)dx=\int f(u)du\) where \(u = g(x)\) and \(du=g^{\prime}(x)dx\).
For the integral \(\int(x^{2}-1)^{12}(2x)dx\), if we let \(u=x^{2}-1\), then \(du = 2x dx\).

Step2: Analyze each option

• Option A: \(2x(x^{2}-1)^{12}\) is the integrand itself, not a substitution variable.
• Option B: \(2(x^{2}-1)^{12}\) does not help in substitution as we need a function whose derivative is also present in the integrand.
• Option C: \((x^{2}-1)^{12}\) is part of the integrand but not the substitution variable.
• Option D: \(x^{2}-1\) is a valid substitution. Let \(u=x^{2}-1\), then \(du = 2x dx\) and the integral \(\int(x^{2}-1)^{12}(2x)dx=\int u^{12}du\) which simplifies the integration process.

Answer:

D. \(x^{2}-1\)