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Chapter 7: Problem 36
Evaluate the following integrals. $$\int \frac{x}{x^{4}+2 x^{2}+1} d x$$
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Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given. $$\int_{0}^{\pi / 2} \ln (\sin x) d x=\int_{0}^{\pi / 2} \ln (\cos x) d x=-\frac{\pi \ln 2}{2}$$
Find the volume of the described solid of revolution or state that it does not exist. The region bounded by \(f(x)=-\ln x\) and the \(x\) -axis on the interval (0,1] is revolved about the \(x\) -axis.
The cycloid is the curve traced by a point on the rim of a rolling wheel. Imagine a wire shaped like an inverted cycloid (see figure). A bead sliding down this wire without friction has some remarkable properties. Among all wire shapes, the cycloid is the shape that produces the fastest descent time (see the Guided Project The amazing cycloid for more about the brachistochrone property). It can be shown that the descent time between any two points \(0 \leq a
Use integration by parts to evaluate the following integrals. $$\int_{0}^{\infty} x e^{-x} d x$$
\(A n\) integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution \(u=\tan (x / 2)\) or equivalently \(x=2 \tan ^{-1} u .\) The following relations are used in making this change of variables. \(A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}\) $$\text { Evaluate } \int_{0}^{\pi / 3} \frac{\sin \theta}{1-\sin \theta} d \theta$$
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