Apentix/AP Calculus AB/Unit 6: Integration and Accumulation of Change/Question

AP Calculus ABhardmcq1 pt

Which of the following equals $\int x\sqrt{2x+1}\,dx$ ?

A.

\frac{1}{10}(2x+1)^{5/2}-\frac{1}{6}(2x+1)^{3/2}+C

B.

\frac{1}{5}(2x+1)^{5/2}-\frac{1}{3}(2x+1)^{3/2}+C

C.

\frac{2}{3}(2x+1)^{3/2}+C

D.

\frac{1}{10}(2x+1)^{5/2}+C

Explanation

Core Concept

Let $u=2x+1$ , so $x=\frac{u-1}{2}$ and $dx=\frac{du}{2}$ . Then $\frac{1}{4}\int(u^{3/2}-u^{1/2})\,du=\frac{1}{10}u^{5/2}-\frac{1}{6}u^{3/2}+C$ .

Correct Answer

A $\frac{1}{10}(2x+1)^{5/2}-\frac{1}{6}(2x+1)^{3/2}+C$

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