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Chapter 6: Problem 36
Find or evaluate the integral. $$ \int \frac{1}{\sec \theta-\tan \theta} d \theta $$
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In your own words, describe how you would integrate \(\int \sin ^{m} x \cos ^{n} x d x\) for each condition. (a) \(m\) is positive and odd. (b) \(n\) is positive and odd. (c) \(m\) and \(n\) are both positive and even.
Graphical Analysis In Exercises 61 and 62, graph \(f(x) / g(x)\) and \(f^{\prime}(x) / g^{\prime}(x)\) near \(x=0 .\) What do you notice about these ratios as \(x \rightarrow 0\) ? How does this illustrate L'H么pital's Rule? \(f(x)=e^{3 x}-1, \quad g(x)=x\)
The Gamma Function \(\Gamma(n)\) is defined by \(\Gamma(n)=\int_{0}^{\infty} x^{n-1} e^{-x} d x, \quad n>0\) (a) Find \(\Gamma(1), \Gamma(2),\) and \(\Gamma(3)\). (b) Use integration by parts to show that \(\Gamma(n+1)=n \Gamma(n)\). (c) Write \(\Gamma(n)\) using factorial notation where \(n\) is a positive integer.
Use mathematical induction to verify that the following integral converges for any positive integer \(n\). \(\int_{0}^{\infty} x^{n} e^{-x} d x\)
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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