7.1E: Exercises

Reading Questions

1. When an integrand contains a radical, such as \(\sqrt[n]{g(x)}\), what is a "radical substitution" you could try?
2. What is the general antiderivative of the function \(f(x) = \frac{1}{a^2+x^2}\)?
3. When integrating \(\int \sin^m(x)\cos^n(x)dx\), what is the general strategy if the power of sine (\(m\)) is odd?
4. What trigonometric identities are required to integrate \(\int \sin^m(x)\cos^n(x)dx\) when both \(m\) and \(n\) are even?
5. When integrating \(\int \tan^k(x)\sec^j(x)dx\), if the power of secant (\(j\)) is even, what is the recommended choice for \(u\) and \(du\)?
6. When integrating \(\int \tan^k(x)\sec^j(x)dx\), if the power of tangent (\(k\)) is odd and there is at least one secant factor, what is the recommended choice for \(u\) and \(du\)?
7. What type of trigonometric identities are used to evaluate integrals like \(\int \sin(5x)\cos(3x)dx\)?

Fill in the blank to make a true statement.

1) \(\sin^2x+\)_______\( =1\)

Answer

\(\cos^2x\)

2) \(\sec^2x−1=\)_______

Answer

\(\tan^2x\)

Use an identity to reduce the power of the trigonometric function to a trigonometric function raised to the first power.

3) \(\sin^2x=\)_______

Answer

\(\dfrac{1−\cos(2x)}{2}\)

4) \(\cos^2x=\)_______

Answer

\(\dfrac{1+\cos(2x)}{2}\)

Evaluate each of the following integrals by \(u\)-substitution.

5) \(\displaystyle \int \sin^3x\cos x\,dx\)

Answer

\(\displaystyle \int \sin^3x\cos x\,dx \quad = \quad \frac{\sin^4x}{4}+C\)

6) \(\displaystyle \int \sqrt{\cos x}\sin x\,dx\)

7) \(\displaystyle \int \tan^5(2x)\sec^2(2x)\,dx\)

Answer

\(\displaystyle \int \tan^5(2x)\sec^2(2x)\,dx \quad = \quad \tfrac{1}{12}\tan^6(2x)+C\)

8) \(\displaystyle \int \sin^7(2x)\cos(2x)\,dx\)

9) \(\displaystyle \int \tan(\frac{x}{2})\sec^2(\frac{x}{2})\,dx\)

Answer

\(\displaystyle \int \tan(\frac{x}{2})\sec^2(\frac{x}{2})\,dx \quad = \quad \tan^2(\frac{x}{2})+C\)

10) \(\displaystyle \int \tan^2x\sec^2x\,dx\)

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)

11) \(\displaystyle \int \sin^3x\,dx\)

Answer

\(\displaystyle \int \sin^3x\,dx \quad = \quad −\frac{3\cos x}{4}+\tfrac{1}{12}\cos(3x)+C=−\cos x+\frac{\cos^3x}{3}+C\)

12) \(\displaystyle \int \cos^3x\,dx\)

13) \(\displaystyle \int \sin x\cos x\,dx\)

Answer

\(\displaystyle \int \sin x\cos x\,dx \quad = \quad −\tfrac{1}{2}\cos^2x+C\)

14) \(\displaystyle \int \cos^5x\,dx\)

15) \(\displaystyle \int \sin^5x\cos^2x\,dx\)

Answer

\(\displaystyle \int \sin^5x\cos^2x\,dx \quad = \quad −\frac{5\cos x}{64}−\tfrac{1}{192}\cos(3x)+\tfrac{3}{320}\cos(5x)−\tfrac{1}{448}\cos(7x)+C\)

16) \(\displaystyle \int \sin^3x\cos^3x\,dx\)

17) \(\displaystyle \int \sqrt{\sin x}\cos x\,dx\)

Answer

\(\displaystyle \int \sqrt{\sin x}\cos x\,dx \quad = \quad \tfrac{2}{3}(\sin x)^{3/2}+C\)

18) \(\displaystyle \int \sqrt{\sin x}\cos^3x\,dx\)

19) \(\displaystyle \int \sec x\tan x\,dx\)

Answer

\(\displaystyle \int \sec x\tan x\,dx \quad = \quad \sec x+C\)

20) \(\displaystyle \int \tan(5x)\,dx\)

21) \(\displaystyle \int \tan x\sec^3x\,dx\)

22) \(\displaystyle \int \sec^4x\,dx\)

Answer

\(\displaystyle \int \sec^4x\,dx \quad = \quad \frac{2\tan x}{3}+\tfrac{1}{3}\sec^2 x\tan x=\tan x+\frac{\tan^3x}{3}+C\)

23) \(\displaystyle \int \cot x\,dx\)

24) \(\displaystyle \int \csc x\,dx\)

Answer

\(\displaystyle \int \csc x\,dx \quad = \quad −\ln|\cot x+\csc x|+C\)

25) \(\displaystyle \int \frac{\tan^3x}{\sqrt{\sec x}}\,dx\)

For exercises 26 - 27, find a general formula for the integrals.

26) \(\displaystyle \int \sin^2ax\cos ax\,dx\)

Answer

\(\displaystyle \int \sin^2ax\cos ax\,dx \quad = \quad \frac{\sin^3(ax)}{3a}+C\)

27) \(\displaystyle \int \sin ax\cos ax\,dx.\)

Use the Double-Angle Identities to evaluate the integrals in exercises 28 - 33.

28) \(\displaystyle \int ^ \pi _0\sin^2x\,dx\)

Answer

\(\displaystyle \int ^ \pi _0\sin^2x\,dx \quad = \quad \frac{ \pi }{2}\)

29) \(\displaystyle \int ^ \pi _0\sin^4 x\,dx\)

30) \(\displaystyle \int \cos^2 3x\,dx\)

Answer

\(\displaystyle \int \cos^2 3x\,dx \quad = \quad \frac{x}{2}+\tfrac{1}{12}\sin(6x)+C\)

31) \(\displaystyle \int \sin^2x\cos^2x\,dx\)

32) \(\displaystyle \int \sin^2x\,dx+ \int \cos^2x\,dx\)

Answer

\(\displaystyle \int \sin^2x\,dx+ \int \cos^2x\,dx \quad = \quad x+C\)

33) \(\displaystyle \int \sin^2 x\cos^2(2x)\,dx\)

For exercises 34 - 42, evaluate the definite integrals. Express answers in exact form whenever possible.

34) \(\displaystyle \int ^{2 \pi }_0\cos x\sin 2x\,dx\)

Answer

\(\displaystyle \int ^{2 \pi }_0\cos x\sin 2x\,dx \quad = \quad 0\)

35) \(\displaystyle \int ^ \pi _0\sin 3x\sin 5x\,dx\)

36) \(\displaystyle \int ^ \pi _0\cos(99x)\sin(101x)\,dx\)

Answer

\(\displaystyle \int ^ \pi _0\cos(99x)\sin(101x)\,dx \quad = \quad 0\)

37) \(\displaystyle \int ^ \pi _{− \pi }\cos^2(3x)\,dx\)

38) \(\displaystyle \int ^{2 \pi }_0\sin x\sin(2x)\sin(3x)\,dx\)

Answer

\(\displaystyle \int ^{2 \pi }_0\sin x\sin(2x)\sin(3x)\,dx \quad = \quad 0\)

39) \(\displaystyle \int ^{4 \pi }_0\cos(x/2)\sin(x/2)\,dx\)

40) \(\displaystyle \int ^{ \pi /3}_{ \pi /6}\frac{\cos^3x}{\sqrt{\sin x}}\,dx\) (Round this answer to three decimal places.)

Answer

\(\displaystyle \int ^{ \pi /3}_{ \pi /6}\frac{\cos^3x}{\sqrt{\sin x}}\,dx \quad \approx \quad 0.239\)

41) \(\displaystyle \int ^{ \pi /3}_{− \pi /3}\sqrt{\sec^2x−1}\,dx\)

42) \(\displaystyle \int ^{ \pi /2}_0\sqrt{1−\cos(2x)}\,dx\)

Answer

\(\displaystyle \int ^{ \pi /2}_0\sqrt{1−\cos(2x)}\,dx \quad = \quad \sqrt{2}\)

43) Find the area of the region bounded by the graphs of the equations \(y=\sin x,\, y=\sin^3x,\, x=0,\) and \(x=\frac{ \pi }{2}.\)

44) Find the area of the region bounded by the graphs of the equations \(y=\cos^2x,\, y=\sin^2x,\, x=−\frac{ \pi }{4},\) and \(x=\frac{ \pi }{4}.\)

Answer

\(A = 1 \,\text{unit}^2\)

45) A particle moves in a straight line with the velocity function \(v(t)=\sin( \omega t)\cos^2( \omega t).\) Find its position function \(x=f(t)\) if \( f(0)=0.\)

46) Find the average value of the function \(f(x)=\sin^2x\cos^3x\) over the interval \([− \pi , \pi ].\)

Answer

\(0\)

For exercises 47 - 48, solve the differential equations.

47) \(\dfrac{dy}{\,dx}=\sin^2x.\) The curve passes through point \((0,0).\)

48) \(\dfrac{dy}{d \theta }=\sin^4( \pi \theta )\)

Answer

\(f(x) = \dfrac{3 \theta }{8}−\tfrac{1}{4 \pi }\sin(2 \pi \theta )+\tfrac{1}{32 \pi }\sin(4 \pi \theta )+C\)

49) Find the length of the curve \(y=\ln(\csc x),\, \text{for}\,\tfrac{ \pi }{4} \leq x \leq \tfrac{ \pi }{2}.\)

50) Find the length of the curve \(y=\ln(\sin x),\, \text{for}\,\tfrac{ \pi }{3} \leq x \leq \tfrac{ \pi }{2}.\)

Answer

\(s = \ln(\sqrt{3})\)

51) Find the volume generated by revolving the curve \(y=\cos(3x)\) about the \(x\)-axis, for \( 0 \leq x \leq \tfrac{ \pi }{36}.\)

For exercises 52 - 53, use this information: The inner product of two functions \(f\) and \(g\) over \([a,b]\) is defined by \(\displaystyle f(x) \cdot g(x)=⟨f,g⟩= \int ^b_af \cdot g\,dx.\) Two distinct functions \(f\) and \(g\) are said to be orthogonal if \(⟨f,g⟩=0.\)

52) Show that \({\sin(2x),\, \cos(3x)}\) are orthogonal over the interval \([− \pi ,\, \pi ]\).

Answer

\(\displaystyle \int ^ \pi _{− \pi }\sin(2x)\cos(3x)\,dx=0\)

53) Evaluate \(\displaystyle \int ^ \pi _{− \pi }\sin(mx)\cos(nx)\,dx.\)

54) Integrate \(y^{\prime}=\sqrt{\tan x}\sec^4x.\)

Answer

\(\displaystyle y = \int \sqrt{\tan x}\sec^4x \, dx \quad = \quad \tfrac{2}{3}\left(\tan x\right)^{3/2} + \tfrac{2}{7}\left(\tan x\right)^{7/2}+C= \tfrac{2}{21}\left(\tan x\right)^{3/2}\left[ 7 + 3\tan^2 x \right]+C\)