# 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
Use the table of integrals at the back of the text to evaluate the integrals $$\displaystyle\int{8}{\sin{{\left({4}{t}\right)}}}{\sin{{\left({\frac{{{t}}}{{{2}}}}\right)}}}{\left.{d}{t}\right.}$$

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,
$$\displaystyle\Rightarrow\int{8}{\sin{{\left({4}{t}\right)}}}{\sin{{\left({\frac{{{t}}}{{{2}}}}\right)}}}{\left.{d}{t}\right.}={4}{\left[{\frac{{{2}}}{{{7}}}}{\sin{{\left({\frac{{{7}{t}}}{{{2}}}}\right)}}}-{\frac{{{2}}}{{{9}}}}{\sin{{\left({\frac{{{9}{t}}}{{{2}}}}\right)}}}\right]}+{C}$$

### Relevant Questions

Evaluate the integral.
$$\displaystyle\int{8}{\sin{{\left({4}{t}\right)}}}{\sin{{\left(\frac{{t}}{{2}}\right)}}}{\left.{d}{t}\right.}$$
Evaluate the integrals using a table of integrals.
$$\displaystyle\int{x}{{\sin}^{{-{{1}}}}{2}}{x}{\left.{d}{x}\right.}$$
Evaluate the integrals.
$$\displaystyle\int{\frac{{{{\sin}^{{-{1}}}{x}}}}{{\sqrt{{{1}-{x}^{{{2}}}}}}}}{\left.{d}{x}\right.}$$
Evaluate the following integrals.
$$\displaystyle\int{{\sin}^{{{2}}}{0}}{{\cos}^{{{5}}}{0}}{d}{0}$$
Trigonometric integral Evaluate the following integrals.
$$\displaystyle\int{{\sin}^{{2}}{0}}{{\cos}^{{5}}{0}}{d}{0}$$
Use a table of integrals to evaluate the definite integral.
$$\displaystyle{\int_{{0}}^{{3}}}\sqrt{{{x}^{{2}}+{16}}}{\left.{d}{x}\right.}$$
$$\displaystyle{\int_{{0}}^{{1}}}{\left[{t}{e}^{{t}}^{2}+{e}^{{-{{t}}}}{j}+{k}\right]}{\left.{d}{t}\right.}$$
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{1}}}^{{{x}}}}{\ln{{t}}}{\left.{d}{t}\right.}$$
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{0}}}^{{{x}}}}{e}^{{\sqrt{{{t}}}}}{\left.{d}{t}\right.}$$
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 $$\displaystyle{m}^{{{3}}}$$ )
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 $$\displaystyle{m}^{{{3}}}$$)