Chapter 8

Calculate $$\int x^{i} \arctan x d x$$.

Find the centroid of the region under the curve \(y=\) \(\left(x^{2}+1\right)^{-1}\) from \(x=0\) to \(x=1\).

The region between the curve \(y=\tan x\) and the \(x\) -axis from \(x=0\) to \(x=\pi / 4\) is revolved about the line \(y=-1 .\) Find the volume of the resulting solid.

Let \(\Omega\) be the region between the graph of the logarithm function and the \(x\) -axis from \(x=1\) to \(x=e\). (a) Find the area of \(\Omega .\) (b) Find the centroid of \(\Omega\). (c) Find the volume of the solids generated by revolving \(\Omega\) about each of the coordinate axis.

Verify that, for each positive integer \(n:\) (a) \(\int_{0}^{\pi} \sin ^{2} n x d x \quad_{2} \pi\) \(\mathrm{HINT}: \sin ^{2} \theta=\frac{1}{2}(1-\cos 2 \theta)\) (b) \(\int_{0}^{x} \sin n x \cos n x d x\) (c) \(\int_{0}^{z / a} \sin n x \cos n x d x=0\)

Let \(\Omega\) be the region under the curve \(y=\sqrt{x^{2}-a^{2}}\) from \(x=a\) to \(x=\sqrt{2} a\) Sketch \(\Omega\), find the area of \(\Omega\), and locate the centroid.

The region between the curve \(y=\sec ^{2} x\) and the \(x\) -axis from \(x=0\) to \(x=\pi / 4\) is revolved about the \(x\) axis. Find the volume of the resulting solid.

Let \(\Omega\) be the region under the curve \(y=\sqrt{x^{2}-a^{2}}\) from \(x=a\) to Find the volume of the solid generated by revolving \(\Omega\) about the \(x\) -axis and determine the centroid of that solid.

(a) Calculate \(\int \sin ^{3} x d x \cdot \operatorname{HINT}: \sin ^{2} x=1-\cos ^{2} x\) (b) Calculate \(\int \sin ^{5} x d x\) (c) Explain how to calculate \(\int \sin ^{2 t-1} x d x\) for an arbitrary positive integer: \(k\)

Let \(f(x)=\frac{\ln x}{x}, x \in[1,2 c]\) (a) Find the ares of the region \(\Omega\) bounded by the graph of \(f\) and the \(x\) -axis. (b) Find the volume of the solid gencrated by revolving \(\Omega\) about the \(x\) -axis.

Use a CAS to decompose into partial fractions. (a) \(\frac{6 x^{4}+11 x^{3}-2 x^{2}-5 x-2}{x^{2}(x+1)^{3}}\). (b) \(-\frac{x^{3}+20 x^{2}+4 x+93}{\left(x^{2}+4\right)\left(x^{2}-9\right)}\). (c) \(\frac{x^{2}+7 x+12}{x\left(x^{2}+2 x+4\right)}\).

(a) Use integration by parts to show that for \(n>2\) $$\int \sin ^{n} x d x=-\frac{1}{n} \sin ^{n-1} x \cos x+\frac{n-1}{n} \int \sin ^{n-2} x d x.$$ (b) Then show that $$\int_{0}^{\pi / 2} \sin ^{\prime \prime} x d x=\frac{n-1}{n} \int_{0}^{\pi / 2} \sin ^{n-2} x d x.$$ (c) Verify the Wallis sine formulas: for even \(n \geq 2\) $$\int_{0}^{\pi / 2} \sin ^{n} x d x=\frac{(n-1) \cdots 5 \cdot 3 \cdot 1}{n \cdots 6 \cdot 4 \cdot 2} \cdot \frac{\pi}{2}.$$ for odd \(n \geq 3,\) $$\int_{0}^{\pi / 2} \sin ^{5} x d x=\frac{(n-1) \cdots 4 \cdot 2}{n \cdots 5 \cdot 3}.$$

Let \(\Omega\) be the region under the curve \(y=\sqrt{x^{2}-a^{2}}\) from \(x=a\) to Find the Find the volume of the solid generated by revolving \(\Omega\) about the \(y\) -axis and determine the centroid of that solid.

Find the centroid of the region under the graph. $$f(x)=e^{x}, \quad x \in[0,1]$$

(a) Calculate \(\int \tan ^{3} x d x \cdot\) HINT: \(\tan ^{2} x=\sec ^{2} x-1\) (b) Calculate \(\int \tan ^{5} x d x\) (c) Calculate \(\int \tan ^{7} x d x\) (d) Explain how to calculate \(\int \tan ^{2 t+1} x d x\) for an arbitrary positive integer \(k\)

Use a CAS to decompose the integrand into partial fractions. Use the decomposition to evaluate the integral.$$\int \frac{2 x^{6}-13 x^{5}+23 x^{4}-15 x^{3}+40 x^{2}-24 x+9}{x^{5}-6 x^{4}+9 x^{2}} d x$$.

Use a trigonometric substitution to derive the formula $$\int \frac{1}{\sqrt{a^{2}+x^{2}}} d x=\ln (x+\sqrt{a^{2}+x^{2}})+C$$

Find the centroid of the region under the graph. $$f(x)=e^{-x}, \quad x \in[0,1]$$

Use a CAS to decompose the integrand into partial fractions. Use the decomposition to evaluate the integral.$$\int \frac{x^{8}+2 x^{7}+7 x^{6}+23 x^{5}+10 x^{4}+95 x^{3}-19 x^{2}+133 x-52}{x^{6}+2 x^{3}+5 x^{4}-16 x^{3}+8 x^{2}+32 x-48} d x$$.

Evaluate by the Wallis formulas: (a) \(\int_{0}^{\pi / 2} \sin ^{7} x d x\). (b) \(\int_{0}^{\pi / 2} \cos ^{6} x d x\).

Use a triponometric substitution to derive the formula.$$\int \frac{1}{\sqrt{x^{2}-a^{2}}} d x=\ln x+\sqrt{x^{2}-a^{2}} |+C$$

Find the centroid of the region under the graph. $$f(x)=\sin x, \quad x \in[0, \pi]$$

(a) Use a graphing utility to sketch the graph of $$f(x)=\frac{1}{\sin x+\cos x} \quad \text { for } \quad 0 \leq x \leq \frac{\pi}{2}$$ (b) Find \(A\) and \(B\) such that \(\sin x+\cos x=A \sin (x+B)\) (c) Find the area of the region between the graph of \(f\) and the \(x\) -axis.

Use a CAS to calculate the integrals $$\int \frac{1}{x^{2}+2 x+n} d x, n=0,1,2$$,Verify your results by differentiation.

Use a graphing utility to draw the graph of the function \(f(x)=x+\sin 2 x, x \in[0, \pi] .\) The region between the graph of \(f\) and the \(x\) -axis is revolved about the \(x\) -axis. (a) Use a CAS to find the volume of the resulting solid. (b) Calculate the volume exactly by carrying out the integration.

$$\text { Set } f(x)=\frac{x^{2}}{\sqrt{1-x^{2}}} \cdot \text { Use a CAS }$$ (a) to draw the graph of \(f\) (b) to find the area between the graph of \(f\) and the \(x\) -axis from \(x=0\) to \(x=\frac{1}{2}\) (c) to find the volume of the solid generated by revolving about the \(y\) -axis the region described in part (b).

Find the centroid of the region under the graph. $$f(x)=\cos x, \quad x \in\left[0, \frac{1}{2} \pi\right]$$

Set $$f(x)=\frac{x}{x^{2}+5 x+6}$$,(a) Use a graphing utility to draw the graph of \(f\) (b) Calculate the area of the region that lies between the graph of \(f\) and the \(x\) -axis from \(x=0\) to \(x=4\).

Use a graphing utility to draw the graph of the function \(g(x)=\sin ^{2} x^{2} \cdot x \subset[0, \sqrt{\pi}] .\) The region between tire graph of \(g\) and the \(x\) -axis is revolved about the \(y\) -axis. (a) Use a CAS to find the volume of the resulting solid. (b) Calculate the volume exactly by carrying the integration.

(a) Use a graphing utility to draw the curves $$y=\frac{x^{2}+1}{x+1} \text { for } x > -1 \text { and } x+2 y=16$$ in the same coordinate system. (b) These curves intersect at two points and determine a bounded region \(\Omega .\) Estimate the \(x\) -coordinates of the two points of intersection accurate to two decimal places. (c) j)determine the approximate area of the region \(\Omega\)

$$\text { Set } f(x)=\frac{\sqrt{x^{2}-9}}{x^{2}}, x \geq 3 . \text { Use a CAS }$$ (a) 10 draw the graph of \(f\) (b) to find the area between the graph of \(f\) and the \(x\) -axis from \(x=3\) to \(x=6\) (c) to locate the centroid of the region described in part (b).

Use a graphing utility to draw in one figure the graphs of both \(f(x)=1+\cos x\) and \(g(x)=\sin _{2} x\) from \(x=0\) to \(x=2 \pi\). (a) Use a CAS to find the points where the two curves is intersect; then find the area between the two curves. (b) The region between the two curves is revolved about the \(x\) -axis. Use a CAS to find the volume of the resulting solid.

The mass density of a rod that extends from \(x=2\) to \(x=3\) is given by the logarithm function \(f(x)=\ln x\). (a) Calculate the mass of the rod. (b) Find the center of mass of the rod.

(a) Use a graphing utility to draw the curve $$y^{2}=x^{2}(1-x)$$ (b) Your drawing \(\mathrm{m}\) part (a) should show that the curve forms a loop for \(0 \leq x \leq 1\). Calculate the area of the loop. HINT: Use the symmetry of the curve.

Find the volume generated by revolving the region under the graph about the \(y\) -axis. $$f(x)=\cos \frac{1}{2} \pi x, \quad x \in[0,1]$$

Find the volume generated by revolving the region under the graph about the \(y\) -axis. $$f(x)=x \sin x, \quad x \in[0, \pi]$$

Find the volume generated by revolving the region under the graph about the \(y\) -axis. $$f(x)=x e^{x}, \quad x \in[0,1]$$

Find the volume generated by revolving the region under the graph about the \(y\) -axis. $$f(x)=x \cos x, \quad x \in\left[0, \frac{1}{2} \pi\right]$$

Let \(\Omega\) be the region under the curve \(y=e^{x}, x \in[0,1] .\) Find the centroid of the solid generated by revolving \(\Omega\) about th: \(x \text { -axis. (For the appropriate formula, sce Project } 6.4 .)\)

Let \(\Omega\) be the region under the graph of \(y=\sin x, x \in\) \(\left[0, \frac{1}{2} \pi\right] .\) Find the centroid of the solid generated by revolving 2 about the \(x\) -axis. (For the appropriate formula, see Praject 6.4.)

Find the volume generated by revolving the region under the graph about the \(y\) -axis. Let \(\Omega\) be the rugion stween the curve \(y=\cosh x\) and the \(x\) axis from \(x=0\) to \(x-1\). Find the area of \(\Omega\) and detcrmine the centroid.

Let \(n\) be a positive integer. Show that $$\int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x$$ The formula given in Exercise 67 reduces the calculation of \(\int x^{n} e^{a x} d x\) to the calculation of \(\int x^{n-1} e^{a r} d x .\) The formula given in Exercise 68 reduces the calculation of \(f(\ln x)^{r} d x\) to the calculation of \(f(\ln x)^{n-1} d x .\) Formulas (such as these) which reduce the calculation of an expression in \(n\) to the calculstion of the corresponding expression in \(n-1\) are called reduction formulas.

Calculate the following integrals by using the appropriate reduction formulas. $$\int x^{3} e^{2 x} d x$$

Calculate the following integrals by using the appropriate reduction formulas. $$\int x^{2} c^{-1} d x$$

Calculate the following integrals by using the appropriate reduction formulas. $$\int(\ln x)^{3} d x$$

Calculate the following integrals by using the appropriate reduction formulas. $$\int(\ln x)^{2} d x$$

(a) As you can probably see, were you to integrate \(\int x^{3} e^{x} d x\) by parts, the result would be of the form $$\int x^{3} e^{x} d x=A x^{3} e^{x}+B x^{2} e^{x}+C x e^{x}+D c^{x}+E$$ Differentiate both sides of this equation and solve for the coefficients \(A, B, C, D .\) In this manner you can calculate the integral without acnially carrying out the integration.

If \(P\) is a polynomial of degree \(k\). then $$\int P(x) e^{x} d x=\left[P(x)-P^{\prime}(x)+\cdots \pm P^{(i)}(x)\right] e^{x}+C$$ Verify this statement. For simplicity, take \(k=4\)

Use the statement in Exercise 74 to calculate: (a) \(\int\left(x^{2}-3 x+1\right) e^{x} d x\) (b) \(\int\left(x^{3}-2 x\right) e^{x} d x\)

Use integration by parts to show that if \(f\) has an inverse with continuous first derivative, then. $$\int f^{-1}(x) d x=x f^{-1}(x)-\int x\left(f^{-1}\right)^{\prime}(x) d x$$