Stewart 8th Section 7.5: Strategy for Integration

89 questions

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Stewart 8th Section 7.5: Strategy for Integration 0/89
1 Mixed integration · Level 1
\( \int \dfrac{\cos x}{1 - \sin x} d x \)
2 Mixed integration · Level 1
\( \displaystyle\int_{0}^{1} (3 x + 1)^{\sqrt{2}} d x \)
3 Mixed integration · Level 2
\( \displaystyle\int_{1}^{4} \sqrt{y} \ln y d y \)
4 Mixed integration · Level 2
\( \int \dfrac{\sin^3 x}{\cos x} d x \)
5 Mixed integration · Level 2
\( \int \dfrac{t}{t^4 + 2} d t \)
6 Mixed integration · Level 2
\( \displaystyle\int_{0}^{1} \dfrac{x}{(2 x + 1)^3} d x \)
7 Mixed integration · Level 2
\( \displaystyle\int_{-1}^1 \dfrac{e^{\arctan y}}{1 + y^2} d y \)
8 Mixed integration · Level 2
\( \int t \sin t \cos t d t \)
9 Mixed integration · Level 2
\( \displaystyle\int_{2}^{4} \dfrac{x + 2}{x^2 + 3 x - 4} d x \)
10 Mixed integration · Level 2
\( \int \dfrac{\cos\left(\dfrac{1}{x}\right)}{x^3} d x \)
11 Mixed integration · Level 3
\( \int \dfrac{1}{x^3 \sqrt{x^2 - 1}} d x \)
12 Mixed integration · Level 3
\( \int \dfrac{2 x - 3}{x^3 + 3 x} d x \)
13 Mixed integration · Level 3
\( \int \sin^5 t \cos^4 t d t \)
14 Mixed integration · Level 2
\( \int \ln(1 + x^2) d x \)
15 Mixed integration · Level 2
\( \int x \sec x \tan x d x \)
16 Mixed integration · Level 3
\( \displaystyle\int_{0}^{\dfrac{\sqrt{2}}{2}} \dfrac{x^2}{\sqrt{1 - x^2}} d x \)
17 Mixed integration · Level 2
\( \displaystyle\int_{0}^{\pi} t \cos^2 t d t \)
18 Mixed integration · Level 2
\( \displaystyle\int_{1}^{4} \dfrac{e^{\sqrt{t}}}{\sqrt{t}} d t \)
19 Mixed integration · Level 2
\( \int e^{x + e^x} d x \)
20 Mixed integration · Level 1
\( \int e^2 d x \)
21 Mixed integration · Level 3
\( \int \arctan \sqrt{x} d x \)
22 Mixed integration · Level 2
\( \int \dfrac{\ln x}{x \sqrt{1 + (\ln x)^2}} d x \)
23 Mixed integration · Level 2
\( \displaystyle\int_{0}^{1} (1 + \sqrt{x})^8 d x \)
24 Mixed integration · Level 2
\( \int (1 + \tan x)^2 \sec x d x \)
25 Mixed integration · Level 2
\( \displaystyle\int_{0}^{1} \dfrac{1 + 12 t}{1 + 3 t} d t \)
26 Mixed integration · Level 3
\( \displaystyle\int_{0}^{1} \dfrac{3 x^2 + 1}{x^3 + x^2 + x + 1} d x \)
27 Mixed integration · Level 2
\( \int \dfrac{d x}{1 + e^x} \)
28 Mixed integration · Level 2
\( \int \sin \sqrt{a t} d t \)
29 Mixed integration · Level 3
\( \int \ln(x + \sqrt{x^2 - 1}) d x \)
30 Mixed integration · Level 2
\( \displaystyle\int_{-1}^2 |e^x - 1| d x \)
31 Mixed integration · Level 3
\( \int \sqrt{\dfrac{1 + x}{1 - x}} d x \)
32 Mixed integration · Level 2
\( \displaystyle\int_{1}^{3} \dfrac{e^{\dfrac{3}{x}}}{x^2} d x \)
33 Mixed integration · Level 3
\( \int \sqrt{3 - 2 x - x^2} d x \)
34 Mixed integration · Level 3
\( \displaystyle\int_{\dfrac{\pi}{4}}^{\dfrac{\pi}{2}} \dfrac{1 + 4 \cot x}{4 - \cot x} d x \)
35 Mixed integration · Level 3
\( \displaystyle\int_{-\dfrac{\pi}{2}}^{\dfrac{\pi}{2}} \dfrac{x}{1 + \cos^2 x} d x \)
36 Mixed integration · Level 3
\( \int \dfrac{1 + \sin x}{1 + \cos x} d x \)
37 Mixed integration · Level 2
\( \displaystyle\int_{0}^{\dfrac{\pi}{4}} \tan^3 \theta \sec^2 \theta d \theta \)
38 Mixed integration · Level 2
\( \displaystyle\int_{\dfrac{\pi}{6}}^{\dfrac{\pi}{3}} \dfrac{\sin \theta \cot \theta}{\sec \theta} d \theta \)
39 Mixed integration · Level 3
\( \int \dfrac{\sec \theta \tan \theta}{\sec^2 \theta - \sec \theta} d \theta \)
40 Mixed integration · Level 2
\( \displaystyle\int_{0}^{\pi} \sin 6 x \cos 3 x d x \)
41 Mixed integration · Level 3
\( \int \theta \tan^2 \theta d \theta \)
42 Mixed integration · Level 3
\( \int \dfrac{\tan^{-1} x}{x^2} d x \)
43 Mixed integration · Level 3
\( \int \dfrac{\sqrt{x}}{1 + x^3} d x \)
44 Mixed integration · Level 3
\( \int \sqrt{1 + e^x} d x \)
45 Mixed integration · Level 3
\( \int x^5 e^{-x^3} d x \)
46 Mixed integration · Level 2
\( \int \dfrac{(x - 1) e^x}{x^2} d x \)
47 Mixed integration · Level 3
\( \int x^3 (x - 1)^{-4} d x \)
48 Mixed integration · Level 3
\( \displaystyle\int_{0}^{1} x \sqrt{2 - \sqrt{1 - x^2}} d x \)
49 Mixed integration · Level 3
\( \int \dfrac{1}{x \sqrt{4 x + 1}} d x \)
50 Mixed integration · Level 3
\( \int \dfrac{1}{x^2 \sqrt{4 x + 1}} d x \)
51 Mixed integration · Level 3
\( \int \dfrac{1}{x \sqrt{4 x^2 + 1}} d x \)
52 Mixed integration · Level 3
\( \int \dfrac{d x}{x(x^4 + 1)} \)
53 Mixed integration · Level 3
\( \int x^2 \sinh m x d x \)
54 Mixed integration · Level 2
\( \int (x + \sin x)^2 d x \)
55 Mixed integration · Level 3
\( \int \dfrac{d x}{x + x \sqrt{x}} \)
56 Mixed integration · Level 3
\( \int \dfrac{d x}{\sqrt{x} + x \sqrt{x}} \)
57 Mixed integration · Level 3
\( \int x \sqrt[3]{x + c} d x \)
58 Mixed integration · Level 3
\( \int \dfrac{x \ln x}{\sqrt{x^2 - 1}} d x \)
59 Mixed integration · Level 3
\( \int \dfrac{d x}{x^4 - 16} \)
60 Mixed integration · Level 3
\( \int \dfrac{d x}{x^2 \sqrt{4 x^2 - 1}} \)
61 Mixed integration · Level 2
\( \int \dfrac{d \theta}{1 + \cos \theta} \)
62 Mixed integration · Level 3
\( \int \dfrac{d \theta}{1 + \cos^2 \theta} \)
63 Mixed integration · Level 3
\( \int \sqrt{x} e^{\sqrt{x}} d x \)
64 Mixed integration · Level 3
\( \int \dfrac{1}{\sqrt{\sqrt{x} + 1}} d x \)
65 Mixed integration · Level 3
\( \int \dfrac{\sin 2 x}{1 + \cos^4 x} d x \)
66 Mixed integration · Level 3
\( \displaystyle\int_{\dfrac{\pi}{4}}^{\dfrac{\pi}{3}} \dfrac{\ln(\tan x)}{\sin x \cos x} d x \)
67 Mixed integration · Level 3
\( \int \dfrac{1}{\sqrt{x + 1} + \sqrt{x}} d x \)
68 Mixed integration · Level 4
\( \int \dfrac{x^2}{x^6 + 3 x^3 + 2} d x \)
69 Mixed integration · Level 3
\( \displaystyle\int_{1}^{\sqrt{3}} \dfrac{\sqrt{1 + x^2}}{x^2} d x \)
70 Mixed integration · Level 4
\( \int \dfrac{1}{1 + 2 e^x - e^{-x}} d x \)
71 Mixed integration · Level 3
\( \int \dfrac{e^{2 x}}{1 + e^x} d x \)
72 Mixed integration · Level 3
\( \int \dfrac{\ln(x + 1)}{x^2} d x \)
73 Mixed integration · Level 2
\( \int \dfrac{x + \arcsin x}{\sqrt{1 - x^2}} d x \)
74 Mixed integration · Level 2
\( \int \dfrac{4^x + 10^x}{2^x} d x \)
75 Mixed integration · Level 3
\( \int \dfrac{d x}{x \ln x - x} \)
76 Mixed integration · Level 3
\( \int \dfrac{x^2}{\sqrt{x^2 + 1}} d x \)
77 Mixed integration · Level 3
\( \int \dfrac{x e^x}{\sqrt{1 + e^x}} d x \)
78 Mixed integration · Level 3
\( \int \dfrac{1 + \sin x}{1 - \sin x} d x \)
79 Mixed integration · Level 3
\( \int x \sin^2 x \cos x d x \)
80 Mixed integration · Level 4
\( \int \dfrac{\sec x \cos 2 x}{\sin x + \sec x} d x \)
81 Mixed integration · Level 3
\( \int \sqrt{1 - \sin x} d x \)
82 Mixed integration · Level 4
\( \int \dfrac{\sin x \cos x}{\sin^4 x + \cos^4 x} d x \)
83 Special antiderivative · Level 3
The functions \(y = e^{x^2}\) and \(y = x^2 e^{x^2}\) don't have elementary antiderivatives, but \(y = (2 x^2 + 1) e^{x^2}\) does. Evaluate \(\int (2 x^2 + 1) e^{x^2} d x\).
84 Expressing in terms of F · Level 4
We know that \(F(x) = \displaystyle\int_{0}^{x} e^{e^t} d t\) is a continuous function by FTC1, though it is not an elementary function. The functions \(\int \dfrac{e^x}{x} d x\) and \(\int \dfrac{1}{\ln x} d x\) are not elementary either, but they can be expressed in terms of \(F\). Evaluate the following integrals in terms of \(F\).
(a) \(\displaystyle\int_{1}^{2} \dfrac{e^x}{x} d x\)
(b) \(\displaystyle\int_{2}^{3} \dfrac{1}{\ln x} d x\)

Enter your answer directly below each part above.

85 Example - Strategy (trigonometric) · Level 3
Evaluate \(\int \dfrac{\tan^3 x}{\cos^3 x} d x\).
86 Example - Strategy (substitution + parts) · Level 2
Evaluate \(\int e^{\sqrt{x}} d x\).
87 Example - Strategy (rational function) · Level 3
Evaluate \(\int \dfrac{x^5 + 1}{x^3 - 3 x^2 - 10 x} d x\).
88 Example - Strategy (substitution) · Level 1
Evaluate \(\int \dfrac{d x}{x \sqrt{\ln x}}\).
89 Example - Strategy (manipulation) · Level 3
Evaluate \(\int \sqrt{\dfrac{1 - x}{1 + x}} d x\).

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