What is the integral of x*cos^2(6*x) ?

The integral of x*cos^2(6*x) is 1/288 (72x^2+12xsin(12x)+cos(12x))+C

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integral of xcos^2(6x)

\int x \cdot cos^{2} (6 \cdot x) d x

\frac{1}{288} (72 x^{2} + 12 x sin (12 x) + cos (12 x)) + C

Solution steps

\int x cos^{2} (6 x) d x

Rewrite using trig identities

= \int \frac{1}{2} x (1 + cos (12 x)) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int x (1 + cos (12 x)) d x

Apply Integration By Parts

= \frac{1}{2} (x (x + \frac{1}{12} sin (12 x)) - \int x + \frac{1}{12} sin (12 x) d x)

\int x + \frac{1}{12} sin (12 x) d x = \frac{x ^{2}}{2} - \frac{1}{144} cos (12 x)

= \frac{1}{2} (x (x + \frac{1}{12} sin (12 x)) - (\frac{x ^{2}}{2} - \frac{1}{144} cos (12 x)))

Simplify

\frac{1}{2} (x (x + \frac{1}{12} sin (12 x)) - (\frac{x ^{2}}{2} - \frac{1}{144} cos (12 x))) : \frac{1}{288} (72 x^{2} + 12 x sin (12 x) + cos (12 x))

= \frac{1}{288} (72 x^{2} + 12 x sin (12 x) + cos (12 x))

Add a constant to the solution

= \frac{1}{288} (72 x^{2} + 12 x sin (12 x) + cos (12 x)) + C

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What is the integral of x*cos^2(6*x) ?
The integral of x*cos^2(6*x) is 1/288 (72x^2+12xsin(12x)+cos(12x))+C