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