91Ó°ÊÓ

Given \(\int_{2}^{6} f(x) d x=10\) and \(\int_{2}^{6} g(x) d x=-2\), evaluate (a) \(\int_{2}^{6}[f(x)+g(x)] d x\). (b) \(\int_{2}^{6}[g(x)-f(x)] d x\). (c) \(\int_{2}^{6} 2 g(x) d x\). (d) \(\int_{2}^{6} 3 f(x) d x\).

Find the indefinite integral. $$ \int \pi \sin \pi x d x $$

Given \(\int_{-1}^{1} f(x) d x=0\) and \(\int_{0}^{1} f(x) d x=5\), evaluate (a) \(\int_{-1}^{0} f(x) d x\). (b) \(\int_{0}^{1} f(x) d x-\int_{-1}^{0} f(x) d x\). (c) \(\int_{-1}^{1} 3 f(x) d x\). (d) \(\int_{0}^{1} 3 f(x) d x\).

Find the indefinite integral. $$ \int 4 x^{3} \sin x^{4} d x $$

Find the value(s) of \(c\) guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. $$ f(x)=2 \sec ^{2} x, \quad[-\pi / 4, \pi / 4] $$

Use the table of values to find lower and upper estimates of \(\int_{0}^{10} f(x) d x\) Assume that \(f\) is a decreasing function. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 32 & 24 & 12 & -4 & -20 & -36 \\ \hline \end{array} $$

Area Use Simpson's Rule with \(n=14\) to approximate the area of the region bounded by the graphs of \(y=\sqrt{x} \cos x\) \(y=0, x=0\), and \(x=\pi / 2\)

Find the indefinite integral. $$ \int \sin 2 x d x $$

Find the value(s) of \(c\) guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. $$ f(x)=\cos x, \quad[-\pi / 3, \pi / 3] $$

Find the indefinite integral. $$ \int \cos 6 x d x $$