Evaluate the integral. int 8 sin(4t) sin(t/2)dt

Evaluate the integral. int 8 sin(4t) sin(t/2)dt

Question
Applications of integrals
asked 2021-03-04
Evaluate the integral.
\(\displaystyle\int{8}{\sin{{\left({4}{t}\right)}}}{\sin{{\left(\frac{{t}}{{2}}\right)}}}{\left.{d}{t}\right.}\)

Answers (1)

2021-03-05
Step 1
Let the given integral is,
\(\displaystyle\int{8}{\sin{{\left({4}{t}\right)}}}{\sin{{\left(\frac{{t}}{{2}}\right)}}}{\left.{d}{t}\right.}\)
By using the formula,
\(\displaystyle{\sin{{\left({a}\right)}}}{\sin{{\left({b}\right)}}}=\frac{{-{\cos{{\left({a}+{b}\right)}}}+{\cos{{\left({a}-{b}\right)}}}}}{{2}}\)
\(\displaystyle\int{8}{\left(\frac{{{\cos{{\left({4}{t}-\frac{{t}}{{2}}\right)}}}-{\cos{{\left({4}{t}+\frac{{t}}{{2}}\right)}}}}}{{2}}\right)}{\left.{d}{t}\right.}\)
\(\displaystyle\Rightarrow{8}\int{\left(\frac{{{\cos{{\left({7}\frac{{t}}{{2}}\right)}}}-{\cos{{\left({9}\frac{{t}}{{2}}\right)}}}}}{{2}}\right)}{\left.{d}{t}\right.}\)
Step 2
By separating the integrals,
\(\displaystyle\Rightarrow\frac{{8}}{{2}}\int{\left({\cos{{\left({7}\frac{{t}}{{2}}\right)}}}\right)}{\left.{d}{t}\right.}-\int{\left({9}\frac{{t}}{{2}}\right)}{)}{\left.{d}{t}\right.}\)
Simplifying this,
\(\displaystyle\Rightarrow\int{8}{\sin{{\left({4}{t}\right)}}}{\sin{{\left(\frac{{t}}{{2}}\right)}}}{\left.{d}{t}\right.}={4}{\left[\frac{{2}}{{7}}{\sin{{\left({7}\frac{{t}}{{2}}\right)}}}-\frac{{2}}{{9}}{\sin{{\left({9}\frac{{t}}{{2}}\right)}}}\right]}+{C}\)
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