What is the integral of cos(7x)x^2 ?

The integral of cos(7x)x^2 is 1/343 (49x^2sin(7x)-2(-7xcos(7x)+sin(7x)))+C

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integral of cos(7x)x^2

\int cos (7 x) x^{2} d x

\frac{1}{343} (49 x^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x))) + C

Solution steps

\int cos (7 x) x^{2} d x

Apply u-substitution

: \int \frac{u ^{2} cos ( u )}{343} d u

= \int \frac{u ^{2} cos ( u )}{343} d u

Take the constant out:

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

= \frac{1}{343} \cdot \int u^{2} cos (u) d u

Apply Integration By Parts

: u^{2} sin (u) - \int 2 u sin (u) d u

= \frac{1}{343} (u^{2} sin (u) - \int 2 u sin (u) d u)

\int 2 u sin (u) d u = 2 (- u cos (u) + sin (u))

= \frac{1}{343} (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

Substitute back

u = 7 x

= \frac{1}{343} ((7 x)^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x)))

Simplify

\frac{1}{343} ((7 x)^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x))) : \frac{1}{343} (49 x^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x)))

= \frac{1}{343} (49 x^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x)))

Add a constant to the solution

= \frac{1}{343} (49 x^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x))) + C

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What is the integral of cos(7x)x^2 ?
The integral of cos(7x)x^2 is 1/343 (49x^2sin(7x)-2(-7xcos(7x)+sin(7x)))+C