Basic Integration Review

Practice Problems with Video Solutions

A Evaluate Each Integral
Watch 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
Watch 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 Definite Integrals

Problem Set 1

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

Problem Set 2

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

Problem Set 3

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

Problem Set 4

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

Problem Set 5

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

Problem Set 6

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

Problem Set 7

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