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Active Calculus 2nd Ed

Matthew Boelkins

Section 9.4 Integration with Trigonometric Functions

Subsection 9.4.1 Preview Activity

Question 9.4.1.

We already know we can do the following problem using u-substitution, with \(u = \sin(9 x )\)
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\(\displaystyle \int \sin^{8}\mathopen{}\left(9x\right)\cos\mathopen{}\left(9x\right) \,dx \ = \ \)
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We see that \(\displaystyle \int \sin^{8} (9 x) \cos^3(9 x) \,dx \ \) is solved similarly, because
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So we’ll still use u-substitution with \(u = \sin(9 x )\text{,}\) after first rewriting \(\cos^3 (9 x) = \cos^2(9 x) \cos(9 x )\) and using a trig identity.
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Select the trigonometric identity needed to solve this problem.
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  • \(\displaystyle \displaystyle 1+ \cot^2 u = \csc^2 u\)
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  • \(\displaystyle \displaystyle \cos^2 u = \frac{1+\cos(2 u)}{2}\)
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  • \(\displaystyle \displaystyle \tan^2 u +1 = \sec^2 u\)
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  • \(\displaystyle \displaystyle \sin^2 u = \frac{1-\cos(2 u)}{2}\)
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  • \(\displaystyle \displaystyle \sin^2 u + \cos^2 u = 1\)
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  • None of the above
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Finally, put it all together to calculate \(\displaystyle \int \sin^{8}\mathopen{}\left(9x\right)\cos^{3}\mathopen{}\left(9x\right) \,dx \ = \ \)
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Hint.
The integral involves \(\sin u\) and \(\cos u\) where the exponent of \(\cos u\) is an odd, positive integer, namely, 3. So we factor out a copy of \(\cos(9 x)\) and use the identity \(\sin^2 u + \cos^2 u = 1\) to turn the remaining \(\cos^2(9 x)\) into an expression involving \(\sin^2(9 x)\text{.}\)
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Thus the integral can be rewritten as
\begin{equation*} \int \sin^{8}\mathopen{}\left(9x\right)\cos^{3}\mathopen{}\left(9x\right) \, dx = \int \sin^{8} (9 x) \cos^2(9 x) \cos(9 x) \, dx \end{equation*}
\begin{equation*} = \int \sin^{8} (9 x) \bigg(1-\sin^2(9 x)\bigg) \cos(9 x) \, dx \end{equation*}
and we use u-substitution.
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Letting \(u= \sin (9 x)\) so \(du = 9 \cos (9 x) \, dx\) and thus \(\displaystyle dx = \frac{ du}{9 \cos (9 x)}\text{.}\)
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This turns the integral into
\begin{equation*} \int u^{8} (1-u^2) \cos(9 x) \, \frac{ du}{9 \cos (9 x)} = \frac{1}{9}\int u^{8} (1-u^2) \, du \end{equation*}
which can be multiplied out, and then each part integrated by power rule.
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Question 9.4.2.

Consider the integral \(\displaystyle \int 10\cos^{2}\mathopen{}\left(2x\right) \,dx\text{.}\)
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In order to integrate, the key feature is that
  • the exponent of \(\sin\) is odd.
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  • the exponent of \(\cos\) is odd.
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  • the exponents of \(\cos\) and \(\sin\) are both even (remember that 0 is even).
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  • None of the above
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Then the trigonometric identity needed to solve this problem is
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  • \(\displaystyle \displaystyle \tan^2 u +1 = \sec^2 u\)
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  • \(\displaystyle \displaystyle \sin^2 u + \cos^2 u = 1\)
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  • \(\displaystyle \displaystyle \cos^2 u = \frac{1+\cos(2 u)}{2}\)
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  • \(\displaystyle \displaystyle \sin^2 u = \frac{1-\cos(2 u)}{2}\)
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  • \(\displaystyle \displaystyle 1+ \cot^2 u = \csc^2 u\)
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  • None of the above
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And we use this identity to rewrite the integral as \(\displaystyle \int 10\cos^{2}\mathopen{}\left(2x\right) \, dx = \ \int\) \(dx\)
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And this can be split into two basic integrals and integrated, though I’m not asking for you to do that here.
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Question 9.4.3.

Select all the key features we could look for to evaluate trig integrals
  • the exponent of \(\tan\) is odd.
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  • the exponent of \(\sin\) is odd.
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  • the exponent of \(\sin\) is even.
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  • the exponent of \(\cos\) is odd.
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  • the exponent of \(\sec\) is even.
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  • the exponent of \(\tan\) is even.
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  • the exponent of \(\sec\) is odd.
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  • the exponent of \(\cos\) is even.
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  • the exponents of both \(\sin\) and \(\cos\) are even.
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  • None of the above
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  • All of the above
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You should be sure to have this information written down so that you can use it to solve problems in class.
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Question 9.4.4.

Consider the integral \(\displaystyle \int 11\tan^{3}\mathopen{}\left(13x\right)\sec^{3}\mathopen{}\left(13x\right) \,dx\text{.}\)
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In order to integrate, the key feature is that
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Then the trigonometric identity needed to solve this problem is
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  • \(\displaystyle \displaystyle \cos^2 u = \frac{1+\cos(2 u)}{2}\)
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    🔗
  • \(\displaystyle \displaystyle 1+ \cot^2 u = \csc^2 u\)
    🔗
    🔗
  • \(\displaystyle \displaystyle \sin^2 u = \frac{1-\cos(2 u)}{2}\)
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  • \(\displaystyle \displaystyle \sin^2 u + \cos^2 u = 1\)
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  • \(\displaystyle \displaystyle \tan^2 u +1 = \sec^2 u\)
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  • None of the above
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So to integrate, we would use u-substitution with \(u= \sec(13 x)\text{.}\) But first we’d have to rewrite the integral in order to have a copy of \(u\)’s derivative \(\sec(13 x) \tan(13 x)\text{,}\) and use a trigonometric identity to turn the remaining even powers of \(\tan(13 x)\) into an expression of \(\sec^2(13 x)\text{.}\)
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Question 9.4.5.

Consider the integral \(\displaystyle \int 5\tan^{4}\mathopen{}\left(15x\right)\sec^{4}\mathopen{}\left(15x\right) \,dx\text{.}\)
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In order to integrate, the key feature is that
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Then the trigonometric identity needed to solve this problem is
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  • \(\displaystyle \displaystyle 1+ \cot^2 u = \csc^2 u\)
    🔗
    🔗
  • \(\displaystyle \displaystyle \sin^2 u + \cos^2 u = 1\)
    🔗
    🔗
  • \(\displaystyle \displaystyle \cos^2 u = \frac{1+\cos(2 u)}{2}\)
    🔗
    🔗
  • \(\displaystyle \displaystyle \sin^2 u = \frac{1-\cos(2 u)}{2}\)
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  • \(\displaystyle \displaystyle \tan^2 u +1 = \sec^2 u\)
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  • None of the above
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So to integrate, we would use u-substitution with \(u= \tan(15 x)\text{.}\) But first we’d have to rewrite the integral in order to have a copy of \(u\)’s derivative \(\sec^2(15 x)\text{,}\) and use a trigonometric identity to turn the remaining even powers of \(\sec(15 x)\) into an expression involving \(\tan^2(15 x)\text{.}\)
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Exercises 9.4.2 Exercises

1.

Evaluate the indefinite integral.
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\(\displaystyle \int 9\tan^{3}\mathopen{}\left(-2x\right)\sec^{3}\mathopen{}\left(-2x\right) \,dx \ = \ \)
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Select the trigonometric identity you used in solving this problem.
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  • \(\displaystyle \displaystyle \tan^2 u +1 = \sec^2 u\)
    🔗
    🔗
  • \(\displaystyle \displaystyle \cos^2 u = \frac{1+\cos(2 u)}{2}\)
    🔗
    🔗
  • \(\displaystyle \displaystyle \sin^2 u + \cos^2 u = 1\)
    🔗
    🔗
  • \(\displaystyle \displaystyle \sin^2 u = \frac{1-\cos(2 u)}{2}\)
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    🔗
  • \(\displaystyle \displaystyle 1+ \cot^2 u = \csc^2 u\)
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  • None of the above
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2.

Evaluate the indefinite integral.
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\(\displaystyle \int 3\tan^{4}\mathopen{}\left(4x\right)\sec^{4}\mathopen{}\left(4x\right) \,dx \ = \ \)
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Select the trigonometric identity you used in solving this problem.
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  • \(\displaystyle \displaystyle \sin^2 u = \frac{1-\cos(2 u)}{2}\)
    🔗
    🔗
  • \(\displaystyle \displaystyle \tan^2 u +1 = \sec^2 u\)
    🔗
    🔗
  • \(\displaystyle \displaystyle 1+ \cot^2 u = \csc^2 u\)
    🔗
    🔗
  • \(\displaystyle \displaystyle \sin^2 u + \cos^2 u = 1\)
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    🔗
  • \(\displaystyle \displaystyle \cos^2 u = \frac{1+\cos(2 u)}{2}\)
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  • None of the above
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3.

Evaluate the indefinite integral.
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\(\displaystyle \int 7\cot^{3}\mathopen{}\left(5x\right)\csc^{3}\mathopen{}\left(5x\right) \,dx \ = \ \)
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Select the trigonometric identity you used in solving this problem.
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  • \(\displaystyle \displaystyle \sin^2 u = \frac{1-\cos(2 u)}{2}\)
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    🔗
  • \(\displaystyle \displaystyle \tan^2 u +1 = \sec^2 u\)
    🔗
    🔗
  • \(\displaystyle \displaystyle \cos^2 u = \frac{1+\cos(2 u)}{2}\)
    🔗
    🔗
  • \(\displaystyle \displaystyle \sin^2 u + \cos^2 u = 1\)
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    🔗
  • \(\displaystyle \displaystyle 1+ \cot^2 u = \csc^2 u\)
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  • None of the above
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4.

Evaluate the indefinite integral.
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\(\displaystyle \int 36 \cos^4(9 x) dx \ =\)
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5.

Evaluate the indefinite integral.
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\(\displaystyle\int \tan^{7}\mathopen{}\left(x\right)\sec^{4}\mathopen{}\left(x\right) \, dx\) =
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6.

Evaluate the integral:
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\(\displaystyle \int {\frac{-7 \tan^3(x)}{\cos^4(x)}}\, dx\ =\)
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7.

Evaluate the integral \(\displaystyle \int {-8 \cot^5(x) \sin^4(x)}\, dx\ =\)
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8.

Evaluate the indefinite integral.
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\(\displaystyle\int \sin(8 x)\cos(16 x)\ dx=\) \(+C\)
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9.

Evaluate the indefinite integral.
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\begin{equation*} \int \sec^3 x \,dx \end{equation*}
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Answer: \(\ + \ C\)
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10.

Match each of the trigonometric expressions below with the equivalent non-trigonometric function from the following list.
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  1. \(\tan(\theta)\) where \(x = 2 \sin \theta\)
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  2. \(\cos(\theta)\) where \(x = 2 \tan \theta\)
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  3. \(\cos(\theta)\) where \(x = 2 \sin \theta\)
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  4. \(\sin(\theta)\) where \(x = 2 \tan \theta\)
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  1. \(\displaystyle \displaystyle \frac{x}{\sqrt{4 - x^2}}\)
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  2. \(\displaystyle \displaystyle \frac{\sqrt{4 - x^2}}{2}\)
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  3. \(\displaystyle \displaystyle \frac{x}{\sqrt{4 + x^2}}\)
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  4. \(\displaystyle \displaystyle \frac{2}{\sqrt{4 + x^2}}\)
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  5. None of the above
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