91Ó°ÊÓ

If necessary, use two or more substitutions to find the following integrals. $$\int_{0}^{1} x \sqrt{1-\sqrt{x}} d x$$

Maximum net area Graph the function \(f(x)=8+2 x-x^{2}\) and determine the values of \(a\) and \(b\) that maximize the value of the integral $$\int_{a}^{b}\left(8+2 x-x^{2}\right) d x$$

If necessary, use two or more substitutions to find the following integrals. $$\int_{0}^{1} \sqrt{x-x \sqrt{x}} d x$$

An integral equation Use the Fundamental Theorem of Calculus, Part 1, to find the function \(f\) that satisfies the equation $$\int_{0}^{x} f(t) d t=2 \cos x+3 x-2$$ Verify the result by substitution into the equation.

Max / min of area functions Suppose \(f\) is continuous on \([0, \infty)\) and \(A(x)\) is the net area of the region bounded by the graph of \(f\) and the \(t\) -axis on \([0, x] .\) Show that the local maxima and minima of \(A\) occur at the zeros of \(f\). Verify this fact with the function \(f(x)=x^{2}-10 x\)

If necessary, use two or more substitutions to find the following integrals. $$\int \tan ^{10} 4 x \sec ^{2} 4 x d x(\text { Hint: Begin with } u=4 x\text { .) }$$

If necessary, use two or more substitutions to find the following integrals. $$\int_{0}^{\pi / 2} \frac{\cos \theta \sin \theta}{\sqrt{\cos ^{2} \theta+16}} d \theta(\text {Hint}: \text { Begin with } u=\cos \theta .)$$

Show that the sine integral \(S(x)=\int_{0}^{x} \frac{\sin t}{t} d t\) satisfies the (differential) equation \(x S^{\prime}(x)+2 S^{\prime \prime}(x)+x S^{\prime \prime \prime}(x)=0\)

Variable integration limits Evaluate \(\frac{d}{d x} \int_{-x}^{x}\left(t^{2}+t\right) d t\) (Hint: Separate the integral into two pieces.)

Discrete version of the Fundamental Theorem In this exercise, we work with a discrete problem and show why the relationship \(\int_{a}^{b} f^{\prime}(x) d x=f(b)-f(a)\) makes sense. Suppose we have a set of equally spaced grid points $$\left\\{a=x_{0}