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

Use the table of integrals at the back of the text to evaluate the integrals

\int 8 \sin (4 t) \sin (\frac{t}{2}) d t

Answer 1

Step 1
Let the given integral is,

\int 8 \sin (4 t) \sin (\frac{t}{2}) d t

By using the formula,

\sin (a) \sin (b) = \frac{- \cos (a + b) + \cos (a - b)}{2}

\int 8 (\frac{\cos (4 t - \frac{t}{2}) - \cos (4 t + \frac{t}{2})}{2}) d t

\Rightarrow 8 \int (\frac{\cos (\frac{7 t}{2}) - \cos (\frac{9 t}{2})}{2}) d t

Step 2
By separating the integrals,

\Rightarrow \frac{8}{2} \int (\cos (\frac{7 t}{2})) d t - \int (\cos (\frac{9 t}{2})) d t

Simplifying this,

\Rightarrow \int 8 \sin (4 t) \sin (\frac{t}{2}) d t = 4 [\frac{2}{7} \sin (\frac{7 t}{2}) - \frac{2}{9} \sin (\frac{9 t}{2})] + C

Answer 2

Jeffrey Jordon

Answered 2021-11-08 Author has 2064 answers

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