Integration (Riemann sums, Fundamental Theorem of Calculus, u-substitution) Knowledge-Based Download PDF

Questions

1. 1.

   What is the value of the definite integral $\int_{0}^{\pi} \sin(x) e^{\cos(x)} \, dx$?

   • A) $e + \frac{1}{e}$
   • B) $e - \frac{1}{e}$
   • C) $e - 1$
   • D) 0
2. 2.

   What is the value of the definite integral $\int_{0}^{2} x^2 \sqrt{x^3 + 1} \, dx$?

   • A) $\frac{52}{3}$
   • B) 6
   • C) $\frac{52}{9}$
   • D) $\frac{26}{9}$
3. 3.

   What is the value of the limit $\lim_{n \to \infty} \sum_{i=1}^{n} \frac{n}{n^2 + i^2}$?

   • A) $\frac{\pi}{4}$
   • B) $\frac{\pi}{2}$
   • C) $\ln(2)$
   • D) $\frac{\pi}{6}$
4. 4.

   What is the value of the definite integral $\int_{1}^{e} \frac{\sqrt{\ln(x)}}{x} \, dx$?

   • A) $\frac{1}{3}$
   • B) $\frac{1}{2}$
   • C) 1
   • D) $\frac{2}{3}$
5. 5.

   If $f(x) = \int_{0}^{x^2} \sqrt{t^2 + 9} \, dt$, what is the value of $f'(2)$?

   • A) 5
   • B) 20
   • C) 10
   • D) 40