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Math 112 Solutions for 5.7, p. 418

1.
(b) $ \frac{3}{7}(1+2^{7/3})$. (d) $ \frac{3}{4}(6^{4/3}-2^{4/3})$. (f) $ (x^{2}+9)^{3/2}+C$. (g) $ \frac{1}{4}(\frac{1}{9}-\frac{1}{225})$. (h) $ \frac{1}{2}\ln(x^{2}+1)+C$. (j) $ \frac{2}{3}(1+x^{3/5})^{3/2}+C$. (l) $ 5^{3/2}-2^{3/2}$.

2.
(b) $ -\frac{1}{3}\cos^{3}x+C$. (c) $ \sin x + \frac{1}{4}\sin
^{4}x+C$. (d) $ \frac{1}{2}$. (f) $ 2\sin
\sqrt x+C$. (g) $ \ln\vert\sin x\vert+C$. (h) $ \ln\vert\sec x\vert+C$. (j) $ -\frac{1}{a}\cos
ax+C$. (l) $ \frac{1}{a}\ln\vert\sec ax\vert+C$.

3.
(b) $ \frac{1}{2}(e^{5}-e^{-1})$. (d) $ -\frac{3}{2}e^{-x^{2}/3}+C$. (f) $ \dfrac{10^{-3x}}{3\ln 10}+C$. (g) $ 2(e^{\sqrt 2}-e)$. (h) $ \ln(1+e^{x})+C$. (j) $ \ln\vert\ln x\vert+C$. (k) $ \frac{1}{4}\cosh^{4}x+C$. (l) $ \tan^{-1}e^{x}+C$.

4.
(b) $ \frac{6}{5}(x+5)^{5/2}-10(x+5)^{3/2}+C$. (d) $ x-5\ln\vert x+5\vert+C$. (f) $ \frac{1}{5}(9-x^{2})^{5/2}-3(9-x^{2})^{3/2}+C$.

5.
(b) $ \int\frac{1}{ax+b}dx=\frac{1}{a}\int\frac{adx}{ax+b}=\frac{1}{a}
\int\frac{dw}{w}=\frac{1}{a}\ln\vert w\vert+C=\frac{1}{a}\ln\vert ax+b\vert+C$.

(c) $ \displaystyle{\int\frac{1}{x^{2}+a^{2}}dx=
\frac{1}{a}\int\frac{(1/a)dx}{1+(\f...
...t\frac{dw}{1+w^{2}}=\frac{1}{a}\tan^{-1}w+C=
\frac{1}{a}\tan^{-1}\frac{x}{a}}+C$.

(d) $ D_{x}(\sin^{-1}\frac{x}{a}+C)=\dfrac{1/a}{\sqrt{1-x^{2}/a^{2}}}=\dfrac{1}{
\sqrt{a^{2}-x^{2}}}$.

6.
To substitute $ u=x^{3}+1$, we must have the derivative, $ x^{2}$, as a factor of the integrand. If $ x^{3}$ or $ x^{4}$ are factors of the integrand, factors of $ x$ or $ x^{2}$ are left over, which we cannot express easily in terms of $ u$.

11.
Use L'H$ \hat{o}$pital.





Jason Grout 2003-04-04