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

Questions

1. 1.

   What is the value of the definite integral $\int_{e}^{e^2} \frac{1}{x \ln(x)} \, dx$?

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

   What is the value of the definite integral $\int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx$?

   • A) $e^2 - e$
   • B) $2(e^2 - e)$
   • C) $2(e^2 - 1)$
   • D) $\frac{1}{2}(e^2 - e)$
3. 3.

   What is the value of the limit $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{\pi}{2n} \sin\left(\frac{k\pi}{2n}\right)$?

   • A) 0
   • B) $\frac{1}{2}$
   • C) $\frac{\pi}{2}$
   • D) 1
4. 4.

   What is the value of the definite integral $\int_0^1 x(1 - x^2)^4 \, dx$?

   • A) $\frac{1}{5}$
   • B) $-\frac{1}{10}$
   • C) $\frac{1}{20}$
   • D) $\frac{1}{10}$
5. 5.

   Let $F(x) = \int_{1}^{x^2} \frac{\cos(t)}{t} \, dt$ for $x > 0$. What is $F'(x)$?

   • A) $\frac{\cos(x^2)}{x^2}$
   • B) $\frac{2\cos(x)}{x}$
   • C) $-\frac{2\sin(x^2)}{x}$
   • D) $\frac{2\cos(x^2)}{x}$

Answers and step-by-step solutions appear after you finish the test.