Chapter 5: Techniques of Integration

If $f$is concave down on all of on [a, b], which of the given approximations is guaranteed to be an over-approximation for ${\int }_a^{b}f\left(x\right)dx$? (Select all that apply.)

(a) left sum

(b) right sum

(c) trapezoid sum

(d) midpoint sum

Use whatever method you like to solve each of the given definite and indefinite integrals. These integrals are neither in order of difficulty nor in order of technique. Many of the integrals can be solved in more than one way.

$\int 2\pi x\left(8-x^{3/2}\right)dx$

Prove the integration formula.

$\int {\sin }^{-1}xdx=x{\sin }^{-1}x+\sqrt{1-x^2}+C$

(a) by applying integration by parts to $\int {\sin }^{-1}xdx$.

(b) by differentiating$\sqrt{1-x^2}+x{\sin }^{-1}x$.

We can extend the technique of trigonometric substitution to the hyperbolic functions. Use Theorem 2.20 and the identity ${\cosh }^{2}x-{\sinh }^{2}x=1$to solve each integral in Exercises 87– 90 with an appropriate hyperbolic substitution $x=a\sinh u\text{oracoshu}$. (These exercises involve hyperbolic functions.)

$\int \frac{x^3}{{\left(x^2-4\right)}^{3/2}}dx$

Use the chain rule and the Fundamental Theorem of Calculus to prove the integration-by-substitution formula for definite integrals:

${\int }_a^{b}f\text{'}\left(u\right(x\left)\right)u\text{'}\left(x\right)dx=f\left(u\right(b\left)\right)-f\left(u\right(a\left)\right)$

Prove the integration formula.

$\int {\tan }^{-1}xdx=x{\tan }^{-1}x-\frac{1}{2}\ln \left(x^2+1\right)+C$

(a) by applying integration by parts to $\int {\tan }^{-1}xdx$.

(a) by differentiating$x{\tan }^{-1}x-\frac{1}{2}\ln \left(x^2+1\right)$.

A pharmaceutical company is designing a new drug whose shape in tablet form is obtained by rotating the graph of ${\left.f\left(x\right)=2\left(4\right)-x^2\right)}^{1/4}$on the interval $[-6,6]$around the x-axis, where units are measured in millimeters.

As we will see in Section 6.1, the volume of such a solid is given by $\pi {\int }_a^b\left(f\right(x){)}^{2}dx$, where a and b are the places where the graph of $f\left(x\right)$meets the x-axis.

(a) Write down a specific definite integral that represents the amount of material, in cubic millimeters, required to make the tablet.

(b) Use trigonometric substitution to solve the definite integral and determine the amount of material needed.

Prove the integration formula

$\int \tan x\mathrm{d}x=\ln \left|\sec x\right|+C$

(a) by using algebra and integration by substitution to find tan x dx;

(b) by differentiating ln | sec x|.

Show that choosing a different anti-derivative $v+C$of $dv$will yield an equivalent formula for integration by parts,

as follows:

(a) Explain why what we need to show is that $uv-\int vdu=u\left(v+C\right)-\int \left(v+C\right)du$.

(b) Rewrite the equation from part (a), substituting$u=u\left(x\right)$, $v=v\left(x\right)$, and $du=u\text{'}\left(x\right)dx$.

(c) Prove the equality you wrote down in part (b).

The main cable on a certain suspension bridge follows a parabolic curve with equation $f(x)=(0.025x{)}^2$, measured in feet, as shown in the following figure:

(a) Write down a specific definite integral that represents the length of the main cable of the suspension bridge.

(b) Use trigonometric substitution to solve the definite integral and determine the length of the cable.