Basic Integration Review

A. Evaluate each integral.   solution

a) \( \displaystyle \int x^3 \, dx \)   b) \( \displaystyle \int e^x \, dx \)   c) \( \displaystyle \int \frac{1}{x} \, dx \)  d) \( \displaystyle \int \sqrt{x} \, dx \)   e) \( \displaystyle \int \sqrt[3]{x^2} \, dx \)   f) \( \displaystyle \int dx \)   g) \( \displaystyle \int 0 \, dx \)   h) \( \displaystyle \int 2 \, dx \)   i) \( \displaystyle \int \pi^2 \, dx \)   j) \( \displaystyle \int e^3 \, dx \)   k) \( \displaystyle \int x^{99} \, dx \)   l) \( \displaystyle \int \frac{1}{x^4} \, dx \)   m) \( \displaystyle \int 3^x \, dx \)   n) \( \displaystyle \int 5 \cdot 2^x \, dx \)

B. Evaluate each integral.   solution

a) \( \displaystyle \int (1 + 2x + e^x) \, dx \)   b) \( \displaystyle \int \frac{x-1}{\sqrt{x}} \, dx \)   c) \( \displaystyle \int (x^7-5x^2+2x-3) \, dx \)   d) \( \displaystyle \int \frac{1+x+x^2}{x} \, dx \)

C. Evaluate the following definite integrals.

1. (a) \( \displaystyle \int_{\pi/4}^{\pi} \sin (\theta) \, d\theta \)   (b) \( \displaystyle \int_0^{\pi/4} 8 \sec^2 (\theta) \, d\theta \)   (c) \( \displaystyle \int_{\pi/6}^{\pi/3} 8 \csc (t) \cot (t) \, dt \)   solution

2. (a) \( \displaystyle \int_0^2 (3x^2-1) \, dx \)   (b) \( \displaystyle \int_{-1}^0 (2x-e^x+1) \, dx \)   (c) \( \displaystyle \int_0^{\pi/4}\frac{1+\cos^2\theta}{\cos^2\theta} \, d\theta \)   solution

3. (a) \( \displaystyle \int_0^1 \frac{2}{1+ t^2} \, dt \)   (b) \( \displaystyle \int_0^9 \sqrt{2x} \, dx \)   solution

4. (a) \( \displaystyle \int_1^{64} \frac{1+\sqrt[3]{x}}{\sqrt{x}} \, dx \)   (b) \( \displaystyle \int_1^9 \frac{x-1}{\sqrt{x}} \, dx \)   solution

5. (a) \( \displaystyle \int_1^{e^2} \frac{5}{x} \, dx \)   (b) \( \displaystyle \int_0^{\ln 5} (e^x-1) \, dx \)   solution

6. (a) \( \displaystyle \int_{-3}^1 e^{v+5} \, dv \)   (b) \( \displaystyle \int_0^2 (6x-3)(8x^2+3) \, dx \)   solution

7. (a) \( \displaystyle \int_0^{\pi/6} \frac{\sqrt{3}\sin(\theta)+ \sqrt{3} \sin(\theta)\tan^2(\theta)}{\sec^2(\theta)} \, d\theta \)   (b) \( \displaystyle \int_0^{\pi/4} \frac{5-5\sin^2(\theta)}{\cos^4(\theta)} \, d\theta \)   solution