5.16 Integration by substitution and by parts Quiz

1. Find $\displaystyle\int x\cos x\,dx$.

• A. $x\sin x - \cos x + C$
• B. $x\sin x + \cos x + C$
• C. $-x\sin x + \cos x + C$
• D. $\dfrac{x^{2}}{2}\sin x + C$

2. For $\displaystyle\int x^{2}\ln x\,dx$, which choice of $u$ is best?

• A. $u = x^{2}$
• B. $u = x$
• C. $u = \ln x$
• D. $u = x^{2}\ln x$

3. Using $u = 2x + 1$, $\displaystyle\int_{0}^{1}(2x + 1)^{3}\,dx$ becomes

• A. $\displaystyle\int_{0}^{1}u^{3}\,du$
• B. $\dfrac{1}{2}\displaystyle\int_{1}^{3}u^{3}\,du$
• C. $2\displaystyle\int_{1}^{3}u^{3}\,du$
• D. $\displaystyle\int_{1}^{3}u^{3}\,du$

4. Find $\displaystyle\int_{1}^{e}\ln x\,dx$.

• A. $e$
• B. $e - 1$
• C. $0$
• D. $1$

5. After two applications of integration by parts to $J = \displaystyle\int e^{x}\cos x\,dx$, you reach $J = e^{x}\sin x + e^{x}\cos x - J$. What is $J$?

• A. $\dfrac{1}{2}e^{x}(\sin x + \cos x) + C$
• B. $e^{x}(\sin x + \cos x) + C$
• C. $\dfrac{1}{2}e^{x}(\sin x - \cos x) + C$
• D. $0$