
# Chapter 7 Review Exercises


For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

1) $$\displaystyle ∫e^xsin(x)dx$$ cannot be integrated by parts.

2) $$\displaystyle ∫\frac{1}{x^4+1}dx$$ cannot be integrated using partial fractions.

Solution: False

3) In numerical integration, increasing the number of points decreases the error.

4) Integration by parts can always yield the integral.

Solution: False

For the following exercises, evaluate the integral using the specified method.

5) $$\displaystyle ∫x^2sin(4x)dx$$ using integration by parts

6) $$\displaystyle ∫\frac{1}{x^2\sqrt{x^2+16}}dx$$ using trigonometric substitution

Solution: $$\displaystyle −\frac{\sqrt{x^2+16}}{16x}+C$$

7) (∫\sqrt{x}ln(x)dx\) using integration by parts

8) $$\displaystyle ∫\frac{3x}{x^3+2x^2−5x−6}dx$$ using partial fractions

Solution: $$\displaystyle \frac{1}{10}(4ln(2−x)+5ln(x+1)−9ln(x+3))+C$$

9) $$\displaystyle ∫\frac{x^5}{(4x^2+4)^{5/2}}dx$$ using trigonometric substitution

10) $$\displaystyle ∫\frac{\sqrt{4−sin^2(x)}}{sin^2(x)}cos(x)dx$$ using a table of integrals or a CAS

Soution: $$\displaystyle −\frac{\sqrt{4−sin^2(x)}}{sin(x)}−\frac{x}{2}+C$$

For the following exercises, integrate using whatever method you choose.

11) $$\displaystyle ∫sin^2(x)cos^2(x)dx$$

12) $$\displaystyle ∫x^3\sqrt{x^2+2}dx$$

Solution: $$\displaystyle \frac{1}{15}(x^2+2)^{3/2}(3x^2−4)+C$$

13) $$\displaystyle ∫\frac{3x^2+1}{x^4−2x^3−x^2+2x}dx$$

14) $$\displaystyle ∫\frac{1}{x^4+4}dx$$

Solution: $$\displaystyle \frac{1}{16}ln(\frac{x^2+2x+2}{x^2−2x+2})−\frac{1}{8}tan^{−1}(1−x)+\frac{1}{8}tan^{−1}(x+1)+C$$

15) $$\displaystyle ∫\frac{\sqrt{3+16x^4}}{x^4}dx$$

For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals.

16) [T] $$\displaystyle ∫^2_1\sqrt{x^5+2}dx$$

Solution: $$\displaystyle M_4=3.312,T_4=3.354,S_4=3.326$$

17) [T] $$\displaystyle ∫^{\sqrt{π}}_0e^{−sin(x^2)}dx$$

18) [T] $$\displaystyle ∫^4_1\frac{ln(1/x)}{x}dx$$

Solution: $$\displaystyle M_4=−0.982,T_4=−0.917,S_4=−0.952$$

For the following exercises, evaluate the integrals, if possible.

19) $$\displaystyle ∫^∞_1\frac{1}{x^n}dx,$$ for what values of $$\displaystyle n$$ does this integral converge or diverge?

20) $$\displaystyle ∫^∞_1\frac{e^{−x}}{x}dx$$

Solution: approximately 0.2194

For the following exercises, consider the gamma function given by $$\displaystyle Γ(a)=∫^∞_0e^{−y}y^{a−1}dy.$$

21) Show that $$\displaystyle Γ(a)=(a−1)Γ(a−1).$$

22) Extend to show that $$\displaystyle Γ(a)=(a−1)!,$$ assuming $$\displaystyle a$$ is a positive integer.

The fastest car in the world, the Bugati Veyron, can reach a top speed of 408 km/h. The graph represents its velocity.

23) [T] Use the graph to estimate the velocity every 20 sec and fit to a graph of the form $$\displaystyle v(t)=aexp^{bx}sin(cx)+d.$$ (Hint: Consider the time units.)

24) [T] Using your function from the previous problem, find exactly how far the Bugati Veyron traveled in the 1 min 40 sec included in the graph.

Answers may vary. Ex: $$\displaystyle 9.405$$ km