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When integrating products of trigonometric functions, the general practice involves applying the trigonometric versions of the Pythagorean Theorem such as
\(\texttt{sin}^2(\theta)+\texttt{cos}^2(\theta)=1\) or \(\texttt{tan}^2(\theta)+1=\texttt{sec}^2(\theta)\)
in conjunction with an appropriate u-substitution. If the powers both even then you may need to consider applying a Power Reduction Identitiy (Half-Angle Idenitty).
\(\texttt{sin}(2\theta)\frac{1-\texttt{cos}(2\theta)}{2}\) or \(\texttt{cos}(2\theta)\frac{1+\texttt{cos}(2\theta)}{2}\)
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Answer the following questions by employing the Pythagorean Theorem or the Power Reduction identities.
Use the double angle identity for sines: \(2\texttt{sin}(x)\texttt{cos}(x) = \texttt{sin}(2x)\), so \(\texttt{sin}(x)\texttt{cos}(x)=\frac{1}{2}\texttt{sin}(2x)\):
\(\normalsize
Apply the power reduction identity for cosines: \(\texttt{cos}^2(x) = \frac{1\:+ \texttt{cos}(2x)}{2}\). Substituting yields:
\(\normalsize
Use a quick U-sub. Let \(u=2x\), so \(du=2dx\). Substituting yields:
\(\normalsize
Apply the power reduction identity for sines in the first integrand: \(\texttt{sin}^2(x) = \frac{1\:- \texttt{cos}(2x)}{2}\). Substituting yields:
\(\normalsize
Use a U-sub for the third integral. Let \(v=\texttt{sin}(u)\), so \(dv=\texttt{cos}(u)du\):
\(\normalsize
Combine previous results into one expression:
\(\normalsizeApply the pythagoras theorem: \(\texttt{sin}^2(x)+\texttt{cos}^2(x) = 1\), so \(\texttt{sin}^2(x) = 1\:- \texttt{cos}^2(x)\)
\(\normalsize
Apply a U-Sub. Let \(u=\texttt{cos}(x)\), so \(du=-\texttt{sin}(x)dx\). Substituting these values leads to:
\(\normalsize
Apply the pythagoras theorem: \(\texttt{sin}^2(x)+\texttt{cos}^2(x) = 1\), so \(\texttt{sin}^2(x) = 1\:- \texttt{cos}^2(x)\)
\(\normalsize
Apply a U-Sub. Let \(u=\texttt{cos}(x)\), so \(du=-\texttt{sin}(x)dx\). Substituting these values leads to:
\(\normalsizeApply the sine and cosine product identity: \(\texttt{sin}(a)\texttt{cos}(b) = \frac{1}{2} \left( \texttt{sin}(a+b) + \texttt{sin}(a-b) \right)\)
\(\normalsize
Apply the pythagoras theorem: \(\texttt{sin}^2(x)+\texttt{cos}^2(x) = 1\), so \(\texttt{sin}^2(x) = 1\:- \texttt{cos}^2(x)\)
\(\normalsize
Apply a U-Sub. Let \(u=\texttt{cos}(x)\), so \(du=-\texttt{sin}(x)dx\). Substituting these values leads to:
\(\normalsizeApply a quick U-sub. Let \(u=4x\), so \(du=4dx\). Substituting these values yield:
\(\normalsize
Apply a quick U-sub. Let \(u=3x\), so \(du=3dx\). Substituting these values yield:
Apply a quick U-sub. Let \(v=\texttt{sec}(u)\), so \(dv=\texttt{sec}(u)\texttt{tan}(u)du\). Substituting these values yield:
Apply a quick U-sub. Let \(u=\texttt{tan}(x)\), so \(du=\texttt{sec}^2(x)dx\). Substituting these values yield:
Use the power reduction identity in order to transform the integrand by parts, first the numerator and then the denominator. Let \(\texttt{sin}^2(x)=\frac{1}{2}[1-\texttt{cos}(2x)]\), so \(2\texttt{sin}^2(x)=1-\texttt{cos}(2x)\). Likewise, let \(\texttt{cos}^2(x)=\frac{1}{2}[1+\texttt{cos}(2x)]\), so \(2\texttt{cos}^2(x)=1+\texttt{cos}(2x)\). Substituting these new values in the integrand yields:
Multiply the integrand by its reciprocal in a fraction:
Perform a quick U-Sub. Let \(u=1\:+ \texttt{cos}(\theta)\), so \(du=-\texttt{sin}(\theta)d\theta\):
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