What is the integral of 125x^2cos(5x) ?

The integral of 125x^2cos(5x) is 25x^2sin(5x)-2(-5xcos(5x)+sin(5x))+C

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integral of 125x^2cos(5x)

\int 125 x^{2} cos (5 x) d x

25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x)) + C

Solution steps

\int 125 x^{2} cos (5 x) d x

Take the constant out:

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

= 125 \cdot \int x^{2} cos (5 x) d x

Apply u-substitution

: \int \frac{u ^{2} cos ( u )}{125} d u

= 125 \cdot \int \frac{u ^{2} cos ( u )}{125} d u

Take the constant out:

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

= 125 \cdot \frac{1}{125} \cdot \int u^{2} cos (u) d u

Apply Integration By Parts

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

= 125 \cdot \frac{1}{125} (u^{2} sin (u) - \int 2 u sin (u) d u)

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

= 125 \cdot \frac{1}{125} (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

Substitute back

u = 5 x

= 125 \cdot \frac{1}{125} ((5 x)^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x)))

Simplify

125 \cdot \frac{1}{125} ((5 x)^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x))) : 25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x))

= 25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x))

Add a constant to the solution

= 25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x)) + C

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What is the integral of 125x^2cos(5x) ?
The integral of 125x^2cos(5x) is 25x^2sin(5x)-2(-5xcos(5x)+sin(5x))+C