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Chapter 7: Problem 45
Evaluate the following integrals or state that they diverge. $$\int_{1}^{\infty} \frac{d x}{\sqrt[3]{x-1}}$$
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When is the volume finite? Let \(R\) be the region bounded by the graph of
\(f(x)=x^{-p}\) and the \(x\) -axis, for \(0
Use the following three identities to evaluate the given integrals. $$\begin{aligned}&\sin m x \sin n x=\frac{1}{2}[\cos ((m-n) x)-\cos ((m+n) x)]\\\&\sin m x \cos n x=\frac{1}{2}[\sin ((m-n) x)+\sin ((m+n) x)]\\\&\cos m x \cos n x=\frac{1}{2}[\cos ((m-n) x)+\cos ((m+n) x)]\end{aligned}$$ $$\int \sin 3 x \sin 2 x d x$$
Find the volume of the following solids. The region bounded by \(y=\frac{1}{\sqrt{4-x^{2}}}, y=0, x=-1,\) ar \(x=1\) is revolved about the \(x\) -axis.
Use the following three identities to evaluate the given integrals. $$\begin{aligned}&\sin m x \sin n x=\frac{1}{2}[\cos ((m-n) x)-\cos ((m+n) x)]\\\&\sin m x \cos n x=\frac{1}{2}[\sin ((m-n) x)+\sin ((m+n) x)]\\\&\cos m x \cos n x=\frac{1}{2}[\cos ((m-n) x)+\cos ((m+n) x)]\end{aligned}$$ $$\int \sin 3 x \cos 7 x d x$$
Decaying oscillations Let \(a>0\) and \(b\) be real numbers. Use integration to confirm the following identities. a. \(\int_{0}^{\infty} e^{-a x} \cos b x d x=\frac{a}{a^{2}+b^{2}}\) b. \(\int_{0}^{\infty} e^{-a x} \sin b x d x=\frac{b}{a^{2}+b^{2}}\)
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