What is the value of the definite integral $\int_0^{\pi/4} x \sec^2(x) \, dx$?

B) $\frac{\pi}{4} - \frac{1}{2}\ln(2)$

C) $\frac{\pi}{4} + \frac{1}{2}\ln(2)$

D) $\frac{\pi}{2} - \frac{1}{2}\ln(2)$

Which of the following is equivalent to the indefinite integral $\int e^{2x} \sin(x) \, dx$?

A) $\frac{e^{2x}}{5} (\sin(x) - 2\cos(x)) + C$

B) $\frac{e^{2x}}{5} (2\sin(x) - \cos(x)) + C$

C) $\frac{e^{2x}}{3} (2\sin(x) - \cos(x)) + C$

D) $\frac{e^{2x}}{5} (2\sin(x) + \cos(x)) + C$

A) $\ln\left(\frac{16}{3}\right)$

B) $\ln\left(\frac{8}{3}\right)$

C) $\ln\left(\frac{16}{9}\right)$

D) $\ln\left(\frac{32}{9}\right)$

A) $-\frac{\ln(x)}{x+1} - \ln(x) + \ln(x+1) + C$

B) $\frac{\ln(x)}{x+1} + \ln(x) - \ln(x+1) + C$

C) $-\frac{\ln(x)}{x+1} + \frac{1}{x+1} + C$

D) $-\frac{\ln(x)}{x+1} + \ln(x) - \ln(x+1) + C$

Math · Calculus · Math · Calculus BC Topics

5 Math practice questions on Integration by parts and partial fractions. Questions that assess core concepts and understanding.

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1.
What is the value of the definite integral $\int_0^{\pi/4} x \sec^2(x) \, dx$ ?
- A) $\frac{\pi}{4} - \ln(2)$
- B) $\frac{\pi}{4} - \frac{1}{2}\ln(2)$
- C) $\frac{\pi}{4} + \frac{1}{2}\ln(2)$
- D) $\frac{\pi}{2} - \frac{1}{2}\ln(2)$
2.
Which of the following is equivalent to the indefinite integral $\int e^{2x} \sin(x) \, dx$ ?
- A) $\frac{e^{2x}}{5} (\sin(x) - 2\cos(x)) + C$
- B) $\frac{e^{2x}}{5} (2\sin(x) - \cos(x)) + C$
- C) $\frac{e^{2x}}{3} (2\sin(x) - \cos(x)) + C$
- D) $\frac{e^{2x}}{5} (2\sin(x) + \cos(x)) + C$
3.
What is the value of the definite integral $\int_0^{\pi/2} x^2 \sin(x) \, dx$ ?
- A) $\pi - 2$
- B) $\pi + 2$
- C) $\frac{\pi^2}{4} - 2$
- D) $2\pi - 1$
4.
What is the value of the definite integral $\int_2^3 \frac{3x - 1}{x^2 - 1} \, dx$ ?
- A) $\ln\left(\frac{16}{3}\right)$
- B) $\ln\left(\frac{8}{3}\right)$
- C) $\ln\left(\frac{16}{9}\right)$
- D) $\ln\left(\frac{32}{9}\right)$
5.
Which of the following is equivalent to the indefinite integral $\int \frac{\ln(x)}{(x+1)^2} \, dx$ for $x > 0$ ?
- A) $-\frac{\ln(x)}{x+1} - \ln(x) + \ln(x+1) + C$
- B) $\frac{\ln(x)}{x+1} + \ln(x) - \ln(x+1) + C$
- C) $-\frac{\ln(x)}{x+1} + \frac{1}{x+1} + C$
- D) $-\frac{\ln(x)}{x+1} + \ln(x) - \ln(x+1) + C$

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