Basic Integration Review
A. Evaluate each integral. solution
a) \( \int x^3 \,
dx\) b) \( \int
e^x \, dx\) c) \( \int
\frac{1}{x} \, dx\)
d) \( \int \sqrt x
dx\) e) \( \int \sqrt[3] {x^2} \,
dx\) f) \( \int dx\)
g) \( \int 0 \,
dx\) h) \( \int 2 \,
dx\) i) \( \int
\pi^2 \, dx\)
j) \( \int e^3 \,
dx\) k) \( \int x^{99}
dx\) l) \( \int \frac{1}{x^4} \, dx\)
m) \( \int 3^x \, dx\) n)
\(\int 5 \cdot 2^x \, dx\)
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. \( \displaystyle{\int_ 1^{64} \frac{1+\sqrt[3]x}{\sqrt{x}} \, dx }\) (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