Chapter 6: Problem 12
In Exercises \(7-12,\) use differentiation to verify the antiderivative formula. $$\int \frac{1}{\sqrt{1-u^{2}}} d u=\sin ^{-1} u+C$$
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In Exercises \(47-52,\) use the given trigonometric identity to set up a \(u\) -substitution and then evaluate the indefinite integral. $$\int 2 \sin ^{2} x d x, \quad \cos 2 x=2 \sin ^{2} x-1$$
Trigonometric Substitution Suppose \(u=\tan ^{-1} x\) (a) Use the substitution \(x=\tan u, d x=\sec ^{2} u d u\) to show that \(\int \frac{d x}{1+x^{2}}=\int 1 d u\) (b) Evaluate \(\int 1 d u\) to show that \(\int \frac{d x}{1+x^{2}}=\tan ^{-1} x+C\)
Multiple Choice \(\int_{0}^{2} e^{2 x} d x=\) (A) \(\frac{e^{4}}{2} \quad(\mathbf{B}) e^{4}-1 \quad\) (C) \(e^{4}-2 \quad\) (D) \(2 e^{4}-2 \quad(\mathbf{E}) \frac{e^{4}-1}{2}\)
In Exercises \(53-66,\) make a \(u\) -substitution and integrate from \(u(a)\) to \(u(b) .\) $$\int_{2}^{5} \frac{d x}{2 x-3}$$
In Exercises \(43-46\) , evaluate the integral by using a substitution prior to integration by parts. $$\int e^{\sqrt{3 x+9}} d x$$
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