Problem 60
derive the given reduction formula using integration by parts. $$ \int \cos ^{\alpha} x d x=\frac{\cos ^{\alpha-1} x \sin x}{\alpha}+\frac{\alpha-1}{\alpha} \int \cos ^{\alpha-2} x d x $$
Problem 61
derive the given reduction formula using integration by parts. $$ \begin{array}{l} \int \cos ^{\alpha} \beta x d x= \\ \quad \frac{\cos ^{\alpha-1} \beta x \sin \beta x}{\alpha \beta}+\frac{\alpha-1}{\alpha} \int \cos ^{\alpha-2} \beta x d x \end{array} $$
Problem 65
. Find the area of the region bounded by the curve \(y=\ln x\), the \(x\) -axis, and the line \(x=e .\)
Problem 67
. Find the area of the region bounded by the curves \(y=3 e^{-x / 3}, y=0, x=0\), and \(x=9 .\) Make a sketch.
Problem 69
Find the area of the region bounded by the graphs of \(y=x \sin x\) and \(y=x \cos x\) from \(x=0\) to \(x=\pi / 4\).
Problem 72
. Evaluate the integral \(\int \cot x \csc ^{2} x d x\) by parts in two different ways: (a) By differentiating \(\cot x\) (b) By differentiating \(\csc x\) (c) Show that the two results are equivalent up to a constant.
Problem 73
If \(p(x)\) is a polynomial of degree \(n\) and \(G_{1}, G_{2}, \ldots, G_{n+1}\), are successive antiderivatives of a function \(g\), then, by repeated integration by parts, \(\int p(x) g(x) d x=p(x) G_{1}(x)-p^{\prime}(x) G_{2}(x)+p^{\prime \prime}(x) G_{3}(x)-\cdots\) \(+(-1)^{n} p^{(n)}(x) G_{n+1}(x)+C\) Use this result to find each of the following: (a) \(\int\left(x^{3}-2 x\right) e^{x} d x\) (b) \(\int\left(x^{2}-3 x+1\right) \sin x d x\)
Problem 77
Find the error in the following "proof" that \(0=1 .\) In \(\int(1 / t) d t\), set \(u=1 / t\) and \(d v=d t .\) Then \(d u=-t^{-2} d t\) and \(u v=1\). Integration by parts gives $$ \int(1 / t) d t=1-\int(-1 / t) d t $$
Problem 78
. Suppose that you want to evaluate the integral $$ \int e^{5 x}(4 \cos 7 x+6 \sin 7 x) d x $$ and you know from experience that the result will be of the form \(e^{5 x}\left(C_{1} \cos 7 x+C_{2} \sin 7 x\right)+C_{3} .\) Compute \(C_{1}\) and \(C_{2}\) by differ- entiating the result and setting it equal to the integrand.
Problem 81
Show that $$ f(t)=f(a)+\sum_{i=1}^{n} \frac{f^{(i)}(a)}{i !}(t-a)^{i}+\int_{a}^{t} \frac{(t-x)^{n}}{n !} f^{(n+1)}(x) d x $$ provided that \(f\) can be differentiated \(n+1\) times.
