Chapter 5: Techniques of Integration

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

As we will see in Section $6.3,$the length of a curve $f\left(x\right)$from $x=a$to $x=b$can be calculated from the formula${\int }_a^b \sqrt{1+{\left(f^{\mathrm{\prime }}\left(x\right)\right)}^2}dx$

(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.

Prove the integration formula

$\int \cot x\mathrm{d}x=-\ln \left|\csc x\right|+C$

(a) by using algebra and integration by substitution to find $\int \cot x\mathrm{d}x$;

(b) by differentiating $-\ln \left|\csc x\right|$.

Prove that the area of a circle of radius $r\text{is}\pi r^2$, as follows:

(a) Write down a definite integral that represents the area of the circle of radius $r$centered at the origin. (Hint: The equation of such a circle is $x^2+y^2=r^2$.)

(b) Use trigonometric substitution to solve the definite integral you found in part (a). (Hint: Change the limits of integration to match your substitution .)

Prove part (b) of Theorem 5.18: For$x\in (-\infty ,\infty )\text{and}u\in (-\pi /2,\pi /2)\text{,}$the substitution $x=a\tan u$ gives $x^2+a^2=a\sec u$. Your proof should include a discussion of domains and a consideration of absolute values.

Factoring polynomials: Factor each of the given polynomials into a product of linear and irreducible quadratic factors. For the third and fourth polynomials you will have to guess a root and use synthetic division or other factoring methods.

a. $x^3-2x^2-x+2$b. $x^3-1$

c. $x^3-4x^2+x+6$ d. $x^3-x^2-5x-3$

Differentiation review: Differentiate each of the functions that follow. Simplify your answers as much as possible.

a. $f\left(x\right)=x{e}^x-e^x$

b. $f\left(x\right)=\frac{x^2\ln x}{2}-\frac{x^2}{4}$

c. $f\left(x\right)=-e^{-x}(x^2+1)-2x{e}^{-x}-2e^{-x}$

d.$f\left(x\right)=\frac{x^3}{3}-2(x{e}^x-e^x)+\frac{e^{2x}}{2}$

e. $f\left(x\right)=-x\cot x+\ln |\sin x|$

f. $f\left(x\right)=x^3\sin x+3x^2\cos x-6x\sin x-6\cos x$

Integration: Calculate the following integrals.

a.$\int \frac{1}{{\left(x-1\right)}^2}dx$b.$\int \frac{x-1}{x^2-1}dx$

c.$\int \frac{1}{x^2+1}dx$d.$\int \frac{x}{x^2+1}dx$

e.$\int \frac{x^2+1}{x^3}dx$f.$\int \frac{x+1}{x^2+1}dx$

Review of integration by substitution: Use u-substitution to find each of the following integrals.

a. $\int e^{3x+1}dx$b. $\int x{e}^{x^2+1}dx$

c. $\int \frac{\ln x}{x}dx$d. $\int \frac{1}{x\ln x}dx$

e. $\int \tan x{\sec }^{2}xdx$f. $\int \frac{1}{x^2+1}dx$

g.$\int \sin x\sqrt{\cos x}dx$h.$\int e^x\sin e^{x}dx$