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

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

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

### Relevant Questions

Evaluate the following integral.
$$\int_{0}^{1}t^{\frac{5}{2}}(\sqrt{t}-3t)dt$$
Evaluate the following integral.
$$\int \frac{3x^{2}+\sqrt{x}}{\sqrt{x}}dx$$
Evaluate the following integral.
$$\int 2x^{3}+3x-2dx$$
Use a change of variables to evaluate the following integral.
$$\int-(\cos^{7}x-5\cos^{5}x-\cos x)\sin x dx$$
Evaluate the following integral: $$\int \frac{(y-3)}{y^{2}-6y+1}$$
Evaluate the following integral: $$\int \frac{x+3}{x-1}dx$$
Evaluate the following integral: $$\int\frac{vdv}{6v^{2}-1}$$
Use the table of integrals at the back of the text to evaluate the integrals $$\displaystyle\int{8}{\sin{{\left({4}{t}\right)}}}{\sin{{\left({\frac{{{t}}}{{{2}}}}\right)}}}{\left.{d}{t}\right.}$$
Find the indefinite integral $$\int \ln(\frac{x}{3})dx$$ (a) using a table of integrals and (b) using the Integration by parts method.
Evaluate the integrals $$\int \cos ec^{4}0d0$$