2.2E: Exercises

Trigonometric Integrals

Fill in the blank to make a true statement.

Exercise \(\PageIndex{1}\)

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

Answer

\(\cos^2x\)

Exercise \(\PageIndex{2}\)

\(\sec^2x−1=\)_______

Answer

\(\tan^2x\)

 

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

Exercise \(\PageIndex{3}\)

\(\sin^2x=\)_______

Answer

\(\dfrac{1−\cos(2x)}{2}\)

Exercise \(\PageIndex{4}\)

\(\cos^2x=\)_______

Answer

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

 

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

Exercise \(\PageIndex{5}\)

\(\displaystyle ∫\sin^3x\cos x\,dx\)

Answer

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

Exercise \(\PageIndex{6}\)

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

Answer

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Exercise \(\PageIndex{7}\)

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

Answer

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

Exercise \(\PageIndex{8}\)

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

Answer

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Exercise \(\PageIndex{9}\)

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

Answer

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

Exercise \(\PageIndex{10}\)

\(\displaystyle ∫\tan^2x\sec^2x\,dx\)

Answer

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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.)

Exercise \(\PageIndex{11}\)

\(\displaystyle ∫\sin^3x\,dx\)

Answer

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

Exercise \(\PageIndex{12}\)

\(\displaystyle ∫\cos^3x\,dx\)

Answer

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Exercise \(\PageIndex{13}\)

\(\displaystyle ∫\sin x\cos x\,dx\)

Answer

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

Exercise \(\PageIndex{14}\)

\(\displaystyle ∫\cos^5x\,dx\)

Answer

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Exercise \(\PageIndex{15}\)

\(\displaystyle ∫\sin^5x\cos^2x\,dx\)

Answer

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

Exercise \(\PageIndex{16}\)

\(\displaystyle ∫\sin^3x\cos^3x\,dx\)

Answer

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Exercise \(\PageIndex{17}\)

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

Answer

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

Exercise \(\PageIndex{18}\)

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

Answer

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Exercise \(\PageIndex{19}\)

\(\displaystyle ∫\sec x\tan x\,dx\)

Answer

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

Exercise \(\PageIndex{20}\)

\(\displaystyle ∫\tan(5x)\,dx\)

Answer

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Exercise \(\PageIndex{21}\)

\(\displaystyle ∫\tan^2x\sec x\,dx\)

Answer

\(\displaystyle ∫\tan^2x\sec x\,dx \quad = \quad \tfrac{1}{2}\sec x\tan x−\tfrac{1}{2}\ln(\sec x+\tan x)+C\)

Exercise \(\PageIndex{22}\)

\(\displaystyle ∫\tan x\sec^3x\,dx\)

Answer

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Exercise \(\PageIndex{23}\)

\(\displaystyle ∫\sec^4x\,dx\)

Answer

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

Exercise \(\PageIndex{24}\)

\(\displaystyle ∫\cot x\,dx\)

Answer

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Exercise \(\PageIndex{25}\)

\(\displaystyle ∫\csc x\,dx\)

Answer

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

Exercise \(\PageIndex{26}\)

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

Answer

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For exercises 27 - 28, find a general formula for the integrals.

Exercise \(\PageIndex{27}\)

\(\displaystyle ∫\sin^2ax\cos ax\,dx\)

Answer

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

Exercise \(\PageIndex{28}\)

\(\displaystyle ∫\sin ax\cos ax\,dx.\)

Answer

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Use the double-angle formulas to evaluate the integrals in exercises 29 - 34.

Exercise \(\PageIndex{29}\)

\(\displaystyle ∫^π_0\sin^2x\,dx\)

Answer

\(\displaystyle ∫^π_0\sin^2x\,dx \quad = \quad \frac{π}{2}\)

Exercise \(\PageIndex{30}\)

\(\displaystyle ∫^π_0\sin^4 x\,dx\)

Answer

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Exercise \(\PageIndex{31}\)

\(\displaystyle ∫\cos^2 3x\,dx\)

Answer

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

Exercise \(\PageIndex{32}\)

\(\displaystyle ∫\sin^2x\cos^2x\,dx\)

Answer

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Exercise \(\PageIndex{33}\)

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

Answer

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

Exercise \(\PageIndex{34}\)

\(\displaystyle ∫\sin^2 x\cos^2(2x)\,dx\)

Answer

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For exercises 35 - 43, evaluate the definite integrals. Express answers in exact form whenever possible.

Exercise \(\PageIndex{35}\)

\(\displaystyle ∫^{2π}_0\cos x\sin 2x\,dx\)

Answer

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

Exercise \(\PageIndex{36}\)

\(\displaystyle ∫^π_0\sin 3x\sin 5x\,dx\)

Answer

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Exercise \(\PageIndex{37}\)

\(\displaystyle ∫^π_0\cos(99x)\sin(101x)\,dx\)

Answer

\(\displaystyle ∫^π_0\cos(99x)\sin(101x)\,dx \quad = \quad 0\)

Exercise \(\PageIndex{38}\)

\(\displaystyle ∫^π_{−π}\cos^2(3x)\,dx\)

Answer

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Exercise \(\PageIndex{39}\)

\(\displaystyle ∫^{2π}_0\sin x\sin(2x)\sin(3x)\,dx\)

Answer

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

Exercise \(\PageIndex{40}\)

\(\displaystyle ∫^{4π}_0\cos(x/2)\sin(x/2)\,dx\)

Answer

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Exercise \(\PageIndex{41}\)

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

Answer

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

Exercise \(\PageIndex{42}\)

\(\displaystyle ∫^{π/4}_{−π/4}\sqrt{\sec^2x−1}\,dx\)

Answer

\( ln(2) \)

 

Solution

\(\displaystyle ∫^{π/4}_{−π/4}\sqrt{\sec^2x−1}\,dx = \displaystyle ∫^{π/4}_{−π/4}\sqrt{\tan^2x}\,dx = \displaystyle ∫^{π/4}_{−π/4} |tan(x)| dx=-  \displaystyle ∫^{0}_{−π/4} tan(x) dx +  \displaystyle ∫^{π/4}_{0} tan(x) dx = ln(2).\)

Exercise \(\PageIndex{43}\)

\(\displaystyle ∫^{π/2}_0\sqrt{1−\cos(2x)}\,dx\)

Answer

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

 

Exercise \(\PageIndex{44}\)

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

Answer

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Exercise \(\PageIndex{45}\)

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

Answer

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

Exercise \(\PageIndex{46}\)

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

Answer

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Exercise \(\PageIndex{47}\)

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

Answer

\(0\)

 

For exercises 48 - 49, solve the differential equations.

Exercise \(\PageIndex{48}\)

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

Answer

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Exercise \(\PageIndex{49}\)

\(\dfrac{dy}{dθ}=\sin^4(πθ)\)

Answer

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

 

Exercise \(\PageIndex{50}\)

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

Answer

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Exercise \(\PageIndex{51}\)

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

Answer

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

Exercise \(\PageIndex{52}\)

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

Answer

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For exercises 53 - 54, use this information: The inner product of two functions \(f\) and \(g\) over \([a,b]\) is defined by \(\displaystyle f(x)⋅g(x)=⟨f,g⟩=∫^b_af⋅g\,dx.\) Two distinct functions \(f\) and \(g\) are said to be orthogonal if \(⟨f,g⟩=0.\)

Exercise \(\PageIndex{53}\)

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

Answer

\(\displaystyle ∫^π_{−π}\sin(2x)\cos(3x)\,dx=0\)

Exercise \(\PageIndex{54}\)

Evaluate \(\displaystyle ∫^π_{−π}\sin(mx)\cos(nx)\,dx.\)

Answer

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Exercise \(\PageIndex{55}\)

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

Answer

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

 

For each pair of integrals in exercises 56 - 57, determine which one is more difficult to evaluate. Explain your reasoning.

Exercise \(\PageIndex{56}\)

\(\displaystyle ∫\sin^{456}x\cos x\,dx\) or \(\displaystyle ∫\sin^2x\cos^2x\,dx\)

Answer

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Exercise \(\PageIndex{57}\)

\(\displaystyle ∫\tan^{350}x\sec^2x\,dx\) or \(\displaystyle ∫\tan^{350}x\sec x\,dx\)

Answer

The second integral is more difficult because the first integral is simply a \(u\)-substitution type.