We have an answer to "Evaluate the integral. ∫−π2π215sin43xcos⁡3xdx"

Wribreeminsl

Answered question

2021-11-08

Evaluate the integral.

\int_{- \frac{π}{2}}^{\frac{π}{2}} 15 \sin^{4} 3 x \cos 3 x d x

nitruraviX

Skilled2021-11-09Added 101 answers

Step 1
The given integral is

\int_{- \frac{π}{2}}^{\frac{π}{2}} 15 \sin^{4} 3 x \cos 3 x d x

.
Step 2
Let u=3x.
Then du=3dx.

\int_{- \frac{π}{2}}^{\frac{π}{2}} 15 \sin^{4} 3 x \cos 3 x d x = 15 \int_{- \frac{π}{2}}^{\frac{π}{2}} \sin^{4} (u) \cos (u) (\frac{d u}{3})

= 5 \int_{1}^{- 1} v^{4} d v

[Assume

v = \sin u \Rightarrow d v = \cos u d u

]

= 5 {[\frac{v^{5}}{5}]}_{1}^{- 1}

= 5 [\frac{{(- 1)}^{5}}{5} - \frac{{(1)}^{5}}{5}]

= 5 [\frac{- 2}{5}]

=-2
Hence, the value of the integral

\int_{- \frac{π}{2}}^{\frac{π}{2}} 15 \sin^{4} 3 x \cos 3 x d x

is -2.

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