Evaluate ∫ sin 4x cos2 4x dx.

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11-04-2018
Mathematics

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Evaluate ∫ sin 4x cos2 4x dx.

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danielmaduroh danielmaduroh · Answer 1 · 2018-05-31T21:46:38+02:00

We have the following integral:

[tex]\int sin(4x)cos(24x)dx[/tex]

Changing variables:

[tex]u=4x \rightarrow dx=\frac{1}{4}dx[/tex]

Therefore:

[tex]\frac{1}{4}\int sin(u)cos(6u)du[/tex]

From trigonometrical identities:

[tex]sin(\alpha)cos(\beta)= \frac{sin(\beta+\alpha)-sin(\beta-\alpha)}{2}[/tex]

Thus:

[tex]\frac{1}{4}\int \frac{sin(7u)-sin(5u)}{2}du= \frac{1}{8} \int sin(7u)du-\frac{1}{8} \int sin(5u)du[/tex]

∴ [tex]\frac{-cos(7u)}{56}+ \frac{cos(5u)}{40}+C[/tex]

[tex]u \rightarrow 4x[/tex]

Finally:

[tex]\boxed{ I= \frac{cos(20x)}{40}-\frac{cos(28x)}{56}+C}[/tex]