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Chapter 6: Problem 18
Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int \frac{x}{\sqrt{2+3 x}} d x $$
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Use integration by parts to verify the reduction formula. $$ \int \sec ^{n} x d x=\frac{1}{n-1} \sec ^{n-2} x \tan x+\frac{n-2}{n-1} \int \sec ^{n-2} x d x $$
Rewrite the improper integral as a proper integral using the given \(u\) -substitution. Then use the Trapezoidal Rule with \(n=5\) to approximate the integral. $$ \int_{0}^{1} \frac{\cos x}{\sqrt{1-x}} d x, u=\sqrt{1-x} $$
(a) find the indefinite integral in two different ways. (b) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (c) Verify analytically that the results differ only by a constant. $$ \int \sec ^{2} x \tan x d x $$
Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=\sin a t $$
The magnetic potential \(P\) at a point on the axis of a circular coil is given by \(P=\frac{2 \pi N I r}{k} \int_{c}^{\infty} \frac{1}{\left(r^{2}+x^{2}\right)^{3 / 2}} d x\) where \(N, I, r, k,\) and \(c\) are constants. Find \(P\)
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