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1
Integration
오답
Evaluate the integral: \(\int \tan^{-1} x d x\)
내 답안
(미작성)
정답
\(x \tan^{-1} x - \dfrac{1}{2} \ln(1 + x^2) + C\)
해설
Integration by parts with \(u = \tan^{-1} x\), \(d v = d x\). Then \(d u = \dfrac{1}{1 + x^2} d x\), \(v = x\).
2
Integration
오답
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int x^n e^{2x^5} d x\)
내 답안
(미작성)
정답
\(n = 4\); \(\dfrac{1}{10}e^{2x^5} + C\)
해설
For \(u = 2x^5\), we need \(d u = 10x^4 d x\), so \(n = 4\). Then \(\int x^4 e^{2x^5} d x = \dfrac{1}{10} \int e^u d u = \dfrac{1}{10}e^{2x^5} + C\).
3
Integration
오답
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int \dfrac{d x}{x^n (\ln x)^7}\)
내 답안
(미작성)
정답
\(n = 1\); \(-\dfrac{1}{6(\ln x)^6} + C\)
해설
For \(u = \ln x\), we need \(d u = \dfrac{1}{x} d x\). So \(n = 1\). Then \(\int \dfrac{d u}{u^7} = -\dfrac{1}{6}u^{-6} + C = -\dfrac{1}{6(\ln x)^6} + C\).
4
Integration
오답
Evaluate the integral: \(\int x^2 \sin x d x\)
내 답안
(미작성)
정답
\(-x^2 \cos x + 2x \sin x + 2 \cos x + C\)
해설
Apply integration by parts twice. First: \(u = x^2\), \(d v = \sin x d x\). Second: \(u = 2x\), \(d v = \cos x d x\).
5
Series
오답
Use the Ratio Test on \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n^2}{2^n}\). The series:
Converges, \(L = \dfrac{1}{2}\)
정답
B
Diverges, \(L = 2\)
C
Inconclusive, \(L = 1\)
D
Converges, \(L = 0\)
해설
\(L = lim |a_{n+1}/a_n| = lim \dfrac{(n+1)^2}{2^{n+1}} \cdot \dfrac{2^n}{n^2} = \dfrac{1}{2} < 1\). Converges.
6
Integration
오답
Find the integer \(n\) that allows for integration by substitution, then evaluate: \(\int x^n e^{-x^2} d x\)
내 답안
(미작성)
정답
\(n = 1\); \(-\dfrac{1}{2}e^{-x^2} + C\)
해설
For \(u = -x^2\), we need \(d u = -2x d x\), so \(n = 1\). Then \(\int x e^{-x^2} d x = -\dfrac{1}{2} \int e^u d u = -\dfrac{1}{2}e^{-x^2} + C\).
7
Integration
오답
Evaluate the integral: \(\int \dfrac{d x}{x(x^2 + 1)}\)
내 답안
(미작성)
정답
\(\ln|x| - \dfrac{1}{2} \ln(x^2 + 1) + C\) or \(\ln|\dfrac{x}{\sqrt{x^2 + 1}}| + C\)
해설
Decompose: \(\dfrac{1}{x(x^2+1)} = \dfrac{A}{x} + \dfrac{B x + C}{x^2 + 1}\). Solve to get \(A = 1\), \(B = -1\), \(C = 0\).
8
Integration
오답
Evaluate the definite integral: \(\displaystyle\int_{-\dfrac{\pi}{2}}^{\dfrac{\pi}{2}} \dfrac{3}{4 + 5 \cos \theta} d \theta\)
내 답안
(미작성)
정답
\(2 \ln 3\)
해설
Use \(t = \tan\left(\dfrac{\theta}{2}\right)\). With \(\cos \theta = \dfrac{1-t^2}{1+t^2}\), the integral transforms and requires careful evaluation of limits as \(\theta \rightarrow \pm \dfrac{\pi}{2}\).
9
Integration
오답
Evaluate the integral: \(\int \sin^{-1} x d x\)
내 답안
(미작성)
정답
\(x \sin^{-1} x + \sqrt{1 - x^2} + C\)
해설
Integration by parts with \(u = \sin^{-1} x\), \(d v = d x\). Then \(d u = \dfrac{1}{\sqrt{1 - x^2}} d x\), \(v = x\).
10
Integration
오답
Evaluate the integral: \(\int \dfrac{x^3 + 2x^2 + 2}{(x^2 + 1)^2} d x\)
내 답안
(미작성)
정답
\(\dfrac{1}{2} \ln(x^2 + 1) + \tan^{-1} x - \dfrac{1}{x^2 + 1} + C\)
해설
Decompose: \(\dfrac{x^3 + 2x^2 + 2}{(x^2+1)^2} = \dfrac{A x + B}{x^2 + 1} + \dfrac{C x + D}{(x^2 + 1)^2}\). The second term requires trig substitution.