Chapter 5

Evaluate the integrals. $$\int x^{3} \sqrt{x^{2}+1} d x$$

Find \(d y / d x\). $$y=\int_{\sqrt{x}}^{0} \sin \left(t^{2}\right) d t$$

Find \(d y / d x\). $$y=x \int_{2}^{x^{2}} \sin \left(t^{3}\right) d t$$

Evaluate the integrals. $$\int 3 x^{5} \sqrt{x^{3}+1} d x$$

Find \(d y / d x\). $$y=\int_{-1}^{x} \frac{t^{2}}{t^{2}+4} d t-\int_{3}^{x} \frac{t^{2}}{t^{2}+4} d t$$

Evaluate the integrals. $$\int \frac{x}{\left(x^{2}-4\right)^{3}} d x$$

Find \(d y / d x\). $$y=\left(\int_{0}^{x}\left(t^{3}+1\right)^{10} d t\right)^{3}$$

Evaluate the integrals. $$\int \frac{x}{(2 x-1)^{2 / 3}} d x$$

Use a definite integral to find the area of the region between the given curve and the \(x\) -axis on the interval \([0, b]\) $$y=3 x^{2}$$

Find \(d y / d x\). $$y=\int_{0}^{\sin x} \frac{d t}{\sqrt{1-t^{2}}}, \quad|x|<\frac{\pi}{2}$$

Evaluate the integrals. $$\int(\cos x) e^{\sin x} d x$$

Find \(d y / d x\). $$y=\int_{\tan x}^{0} \frac{d t}{1+t^{2}}$$

Evaluate the integrals. $$\int(\sin 2 \theta) e^{\sin ^{2} \theta} d \theta$$

Find \(d y / d x\). $$y=\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}} d t$$

Evaluate the integrals. $$\int \frac{1}{\sqrt{x} e^{-\sqrt{x}}} \sec ^{2}\left(e^{\sqrt{x}}+1\right) d x$$

Use a definite integral to find the area of the region between the given curve and the \(x\) -axis on the interval \([0, b]\) $$y=\frac{x}{2}+1$$

Find \(d y / d x\). $$y=\int_{2^{x}}^{1} \sqrt[3]{t} d t$$

Evaluate the integrals. $$\int \frac{1}{x^{2}} e^{1 / x} \sec \left(1+e^{1 / x}\right) \tan \left(1+e^{1 / x}\right) d x$$

Graph the function and find its average value over the given interval. $$f(x)=x^{2}-1 \text { on }[0, \sqrt{3}]$$

Find \(d y / d x\). $$y=\int_{0}^{\sin ^{-1} x} \cos t d t$$

Evaluate the integrals. $$\int \frac{d x}{x \ln x}$$

Graph the function and find its average value over the given interval. $$f(x)=-\frac{x^{2}}{2} \quad \text { on } \quad[0,3]$$

Find \(d y / d x\). $$y=\int_{-1}^{x^{1 / 2}} \sin ^{-1} t d t$$

Evaluate the integrals. $$\int \frac{\ln \sqrt{t}}{t} d t$$

Find the total area between the region and the \(x\) -axis. $$y=-x^{2}-2 x,-3 \leq x \leq 2$$

Graph the function and find its average value over the given interval. $$f(x)=-3 x^{2}-1 \quad \text { on } \quad[0,1]$$

Evaluate the integrals. $$\int \frac{d z}{1+e^{z}}$$

Find the total area between the region and the \(x\) -axis. $$y=3 x^{2}-3, \quad-2 \leq x \leq 2$$

Graph the function and find its average value over the given interval. $$f(x)=3 x^{2}-3 \text { on }[0,1]$$

Evaluate the integrals. $$\int \frac{d x}{x \sqrt{x^{4}-1}}$$

Find the total area between the region and the \(x\) -axis. $$y=x^{3}-3 x^{2}+2 x, \quad 0 \leq x \leq 2$$

Graph the function and find its average value over the given interval. $$f(t)=(t-1)^{2} \quad \text { on } \quad[0,3]$$

Evaluate the integrals. $$\int \frac{5}{9+4 r^{2}} d r$$

Find the total area between the region and the \(x\) -axis. $$y=x^{1 / 3}-x, \quad-1 \leq x \leq 8$$

Graph the function and find its average value over the given interval. $$f(t)=t^{2}-t \quad \text { on } \quad[-2,1]$$

Evaluate the integrals. $$\int \frac{1}{\sqrt{e^{2 \theta}-1}} d \theta$$

Graph the function and find its average value over the given interval. $$g(x)=|x|-1 \text { on } \mathbf{a} .[-1,1], \text { b. }[1,3], \text { and } \mathbf{c} .[-1,3]$$

Evaluate the integrals. $$\int \frac{e^{\sin ^{-1} x} d x}{\sqrt{1-x^{2}}}$$

Graph the function and find its average value over the given interval. $$h(x)=-|x| \quad \text { on } \quad \text { a. }[-1,0], \text { b. }[0,1], \text { and } c .[-1,1]$$

Evaluate the integrals. $$\int \frac{e^{\cos ^{-1} x} d x}{\sqrt{1-x^{2}}}$$

Evaluate the integrals. $$\int \frac{\left(\sin ^{-1} x\right)^{2} d x}{\sqrt{1-x^{2}}}$$

Find the areas of the regions enclosed by the lines and curves in Exercises \(63-72\). $$y=2 x-x^{2} \quad \text { and } \quad y=-3$$

Evaluate the integrals. $$\int \frac{\sqrt{\tan ^{-1} x} d x}{1+x^{2}}$$

Evaluate the integrals. $$\int \frac{d y}{\left(\tan ^{-1} y\right)\left(1+y^{2}\right)}$$

Each of the following functions solves one of the initial value problems.Which function solves which problem? Give brief reasons for your answers. a. \(y=\int_{1}^{x} \frac{1}{t} d t-3\) b. \(y=\int_{0}^{x} \sec t d t+4\) c. \(y=\int_{-1}^{x} \sec t d t+4\) d. \(y=\int_{\pi}^{x} \frac{1}{t} d t-3\) $$y^{\prime}=\sec x, \quad y(-1)=4$$

Find the areas of the regions enclosed by the lines and curves in Exercises \(63-72\). $$y=x^{2}-2 x \quad \text { and } \quad y=x$$

Evaluate the integrals. $$\int \frac{d y}{\left(\sin ^{-1} y\right) \sqrt{1-y^{2}}}$$

If you do not know what substitution to make, try reducing the integral step by step, using a trial substitution to simplify the integral a bit and then another to simplify it some more. You will see what we mean if you try the sequences of substitutions. \(\int \frac{18 \tan ^{2} x \sec ^{2} x}{\left(2+\tan ^{3} x\right)^{2}} d x\) a. \(u=\tan x,\) followed by \(v=u^{3},\) then by \(w=2+v\) b. \(u=\tan ^{3} x,\) followed by \(v=2+u\) c. \(u=2+\tan ^{3} x\)

Each of the following functions solves one of the initial value problems.Which function solves which problem? Give brief reasons for your answers. a. \(y=\int_{1}^{x} \frac{1}{t} d t-3\) b. \(y=\int_{0}^{x} \sec t d t+4\) c. \(y=\int_{-1}^{x} \sec t d t+4\) d. \(y=\int_{\pi}^{x} \frac{1}{t} d t-3\) $$y^{\prime}=\frac{1}{x}, \quad y(1)=-3$$

Find the areas of the regions enclosed by the lines and curves in Exercises \(63-72\). $$y=7-2 x^{2} \quad \text { and } \quad y=x^{2}+4$$