Integration by parts and partial fractions Knowledge-Based Download PDF

Questions

1. 1.

   Evaluate the indefinite integral $\int e^{2x} \cos x \, dx$.

   • A) $\frac{e^{2x}}{5}(2\sin x - \cos x) + C$
   • B) $\frac{e^{2x}}{3}(\sin x + 2\cos x) + C$
   • C) $\frac{e^{2x}}{5}(\sin x + 2\cos x) + C$
   • D) $\frac{e^{2x}}{5}(2\cos x - \sin x) + C$
2. 2.

   Evaluate the indefinite integral $\int \frac{x^3 - 2x}{x^2 + 3x + 2} \, dx$ for $x > 0$.

   • A) $\frac{1}{2}x^2 + 3x + \ln(x+1) + 4\ln(x+2) + C$
   • B) $\frac{1}{2}x^2 - 3x + 4\ln(x+1) + \ln(x+2) + C$
   • C) $\frac{1}{2}x^2 - 3x + \ln(x+1) + 4\ln(x+2) + C$
   • D) $x^2 - 3x + 2\ln(x+1) + 3\ln(x+2) + C$
3. 3.

   Evaluate the indefinite integral $\int \frac{\arctan x}{x^2} \, dx$ for $x > 0$.

   • A) $-\frac{\arctan x}{x} - \ln x + \frac{1}{2} \ln(x^2 + 1) + C$
   • B) $\frac{\arctan x}{x} + \ln x - \frac{1}{2} \ln(x^2 + 1) + C$
   • C) $-\frac{\arctan x}{x} + \ln x - \ln(x^2 + 1) + C$
   • D) $-\frac{\arctan x}{x} + \ln x - \frac{1}{2} \ln(x^2 + 1) + C$
4. 4.

   Which of the following is equivalent to the indefinite integral $\int \ln(x^2 - 1) \, dx$ for $x > 1$?

   • A) $(x - 1)\ln(x - 1) + (x + 1)\ln(x + 1) - 2x + C$
   • B) $(x + 1)\ln(x - 1) + (x - 1)\ln(x + 1) - 2x + C$
   • C) $(x - 1)\ln(x - 1) - (x + 1)\ln(x + 1) + 2x + C$
   • D) $x\ln(x^2 - 1) - x + C$
5. 5.

   Evaluate 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) $-\frac{\ln x}{(x+1)^2} + \ln x - \ln(x+1) + C$
   • D) $-\frac{\ln x}{x+1} + \ln x - \ln(x+1) + C$