Use the table of integrals at the back of the text to evaluate the integrals \int 8\sin(4t)\sin(\frac{t}{2})dt

Use the table of integrals at the back of the text to evaluate the integrals \int 8\sin(4t)\sin(\frac{t}{2})dt

Question
Applications of integrals
asked 2020-12-28
Use the table of integrals at the back of the text to evaluate the integrals \(\displaystyle\int{8}{\sin{{\left({4}{t}\right)}}}{\sin{{\left({\frac{{{t}}}{{{2}}}}\right)}}}{\left.{d}{t}\right.}\)

Answers (1)

2020-12-29
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,
\(\displaystyle{\sin{{\left({a}\right)}}}{\sin{{\left({b}\right)}}}={\frac{{-{\cos{{\left({a}+{b}\right)}}}+{\cos{{\left({a}-{b}\right)}}}}}{{{2}}}}\)
\(\displaystyle\int{8}{\left({\frac{{{\cos{{\left({4}{t}-{\frac{{{t}}}{{{2}}}}\right)}}}-{\cos{{\left({4}{t}+{\frac{{{t}}}{{{2}}}}\right)}}}}}{{{2}}}}\right)}{\left.{d}{t}\right.}\)
\(\displaystyle\Rightarrow{8}\int{\left({\frac{{{\cos{{\left({\frac{{{7}{t}}}{{{2}}}}\right)}}}-{\cos{{\left({\frac{{{9}{t}}}{{{2}}}}\right)}}}}}{{{2}}}}\right)}{\left.{d}{t}\right.}\)
Step 2
By separating the integrals,
\(\displaystyle\Rightarrow{\frac{{{8}}}{{{2}}}}\int{\left({\cos{{\left({\frac{{{7}{t}}}{{{2}}}}\right)}}}\right)}{\left.{d}{t}\right.}-\int{\left({\cos{{\left({\frac{{{9}{t}}}{{{2}}}}\right)}}}\right)}{\left.{d}{t}\right.}\)
Simplifying this,
\(\displaystyle\Rightarrow\int{8}{\sin{{\left({4}{t}\right)}}}{\sin{{\left({\frac{{{t}}}{{{2}}}}\right)}}}{\left.{d}{t}\right.}={4}{\left[{\frac{{{2}}}{{{7}}}}{\sin{{\left({\frac{{{7}{t}}}{{{2}}}}\right)}}}-{\frac{{{2}}}{{{9}}}}{\sin{{\left({\frac{{{9}{t}}}{{{2}}}}\right)}}}\right]}+{C}\)
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Show transcribed image text A log 10 m long is cut at 1 meter intervals and itscross-sectional areas A (at a distance x from theend of the log) are listed in the table. Use the Midpoint Rule withn = 5 to estimate the volume of the log. (in \(\displaystyle{m}^{{{3}}}\))
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