What is the integral of pixcos(2x) ?

The integral of pixcos(2x) is pi/4 (2xsin(2x)+cos(2x))+C

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integral of pixcos(2x)

\int π x cos (2 x) d x

\frac{π}{4} (2 x sin (2 x) + cos (2 x)) + C

Solution steps

\int π x cos (2 x) d x

Take the constant out:

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

= π \cdot \int x cos (2 x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{4} d u

= π \cdot \int \frac{u cos ( u )}{4} d u

Take the constant out:

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

= π \frac{1}{4} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= π \frac{1}{4} (u sin (u) - \int sin (u) d u)

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

= π \frac{1}{4} (u sin (u) - (- cos (u)))

Substitute back

u = 2 x

= π \frac{1}{4} (2 x sin (2 x) - (- cos (2 x)))

Simplify

π \frac{1}{4} (2 x sin (2 x) - (- cos (2 x))) : \frac{π}{4} (2 x sin (2 x) + cos (2 x))

= \frac{π}{4} (2 x sin (2 x) + cos (2 x))

Add a constant to the solution

= \frac{π}{4} (2 x sin (2 x) + cos (2 x)) + C

n = 1 \sum \infty cos (\frac{1}{n ^{3}})

\int 15 x^{2} (x^{3} - 1)^{4} d x

- 15 x^{4} + 270 x^{2}

\frac{d}{d x} (5 e^{6 x})

\int 8 x^{3} e^{- x^{2}} d x

What is the integral of pixcos(2x) ?
The integral of pixcos(2x) is pi/4 (2xsin(2x)+cos(2x))+C