1
Evaluate the integral: \(\int x^3 \sqrt{4 + x^4} d x\)
2
Evaluate the integral: \(\int \dfrac{d x}{x \ln x}\)
3
Evaluate the integral: \(\int \dfrac{(x + 5) d x}{\sqrt{x + 4}}\)
4
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int x^n \sqrt{1 - x^4} d x\)
5
Find the integer \(n\) that allows for integration by substitution (two natural choices exist), then evaluate: \(\int \dfrac{x^n}{\sqrt{1 - x^4}} d x\)
6
Find the integer \(n\) that allows for integration by substitution (two natural choices exist), then evaluate: \(\int \dfrac{x^n}{1 + x^10} d x\)
7
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int \dfrac{x^6}{1 + x^n} d x\)
8
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int x^n e^{-x^2} d x\)
9
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int x^n e^{2x^5} d x\)
10
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int x^5 \sqrt{1 - x^n} d x\)
11
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int \dfrac{x^6}{\sqrt{1 - x^n}} d x\)
12
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int \dfrac{d x}{x^n \ln x}\)
13
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int \dfrac{d x}{x^n (\ln x)^7}\)
14
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int x^n \sin(x^6) d x\)
15
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int \dfrac{\sin^n x \cos x}{\sqrt{3 + \sin^4 x}} d x\)
16
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int \dfrac{\sin^3 x \cos x}{\sqrt{3 + \sin^n x}} d x\)
17
Evaluate the integral: \(\int x e^{-\dfrac{x}{10}} d x\)
18
Evaluate the integral: \(\int x^2 e^{-\dfrac{x}{10}} d x\)
19
Evaluate the integral: \(\int x^2 \ln x d x\)
20
Evaluate the integral for integer \(n \neq -1\): \(\int x^n \ln x d x\)
21
Evaluate the integral: \(\int x^2 \sin x d x\)
22
Evaluate the integral: \(\int x^3 e^{-x^2} d x\)
23
Evaluate the integral: \(\int x^3 \sqrt{x^2 + 1} d x\)
24
Given that \(\int f(x) d x = g(x)\) and \(\int g(x) d x = h(x)\), compute: \(\int x^3 f(x^2) d x\)
25
Given that \(\int f(x) d x = g(x)\) and \(\int g(x) d x = h(x)\), compute: \(\int x^{2n-1} f(x^n) d x\)
26
Evaluate the integral: \(\int \sin^{-1} x d x\)
27
Evaluate the integral: \(\int (\sin^{-1} x)^2 d x\)
28
Evaluate the integral: \(\int \tan^{-1} x d x\)
29
Evaluate the integral: \(\int \sec^3 \theta d \theta\)
Hint: Write \(\sec^3 \theta = \sec \theta (1 + \tan^2 \theta)\) and integrate \(\sec \theta \tan^2 \theta\) by parts.
30
Evaluate the integral: \(\int \dfrac{\sqrt{9 - x^2}}{x^2} d x\)
31
Evaluate the integral: \(\int \dfrac{d x}{x \sqrt{1 - x^2}}\)
32
Evaluate the integral: \(\int \dfrac{d x}{x \sqrt{a^2 + x^2}}\)
33
Evaluate the integral: \(\int \sqrt{4 + x^2} d x\)
Hint: See Problem 12 on page 3 (integral of \(\sec^3 \theta\)).
34
Evaluate the integral: \(\int \dfrac{d x}{a^2 - x^2}\)
Note: It might be easier to do this by partial fractions.
35
Evaluate the integral: \(\int \dfrac{\sqrt{x^2 - a^2}}{x} d x\)
36
Evaluate the integral: \(\int \dfrac{d x}{(a^2 + x^2)^2}\)
37
Evaluate the integral using the substitution \(x = \sin \theta\): \(\int \sin^{-1} x d x\)
38
Evaluate the integral: \(\int (\sin^{-1} x)^2 d x\)
39
Evaluate the integral: \(\int \tan^{-1} x d x\)
40
Evaluate the integral: \(\int \dfrac{5x - 3}{x^2 - 2x - 3} d x\)
41
Evaluate the integral: \(\int \dfrac{6x + 7}{(x + 2)^2} d x\)
42
Evaluate the integral: \(\int \dfrac{2x^3 - 4x^2 - x - 3}{x^2 - 2x - 3} d x\)
43
Evaluate the integral: \(\int \dfrac{d x}{x(x^2 + 1)}\)
44
Evaluate the integral: \(\int \left(\dfrac{1}{x^2 + 1} - \dfrac{1}{x^2 - 2x + 5}\right) d x\)
45
Evaluate the integral: \(\int \dfrac{x^3 + 2x^2 + 2}{(x^2 + 1)^2} d x\)
46
Evaluate the definite integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \dfrac{3}{1 + \sin \theta} d \theta\)
47
Evaluate the definite integral: \(\displaystyle\int_{0}^{2 \dfrac{\pi}{3}} \dfrac{3}{5 + 4 \cos \theta} d \theta\)
48
Evaluate the definite integral: \(\displaystyle\int_{-\dfrac{\pi}{2}}^{\dfrac{\pi}{2}} \dfrac{3}{4 + 5 \cos \theta} d \theta\)
49
Evaluate the definite integral: \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \dfrac{5}{3 \sin \theta + 4 \cos \theta} d \theta\)
50
Solve the initial value problem: \(\dfrac{d y}{d x} = x y\), \(y(0) = 1\)
51
Solve the initial value problem: \(y \dfrac{d y}{d x} = x^2\), \(y(0) = 1\)
52
Solve the initial value problem: \(\dfrac{d y}{d x} = -2x(y + 3)\), \(y(0) = 1\)
53
Solve the initial value problem: \(\dfrac{d y}{d x} = \dfrac{x^2 y + y}{x^2 - 1}\), \(y(0) = 2\)
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