What is the integral of 2x^2cos(2pix^3) ?

The integral of 2x^2cos(2pix^3) is 1/(3pi)sin(2pix^3)+C

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integral of 2x^2cos(2pix^3)

\int 2 x^{2} cos (2 π x^{3}) d x

\frac{1}{3 π} sin (2 π x^{3}) + C

Solution steps

\int 2 x^{2} cos (2 π x^{3}) d x

Take the constant out:

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

= 2 \cdot \int x^{2} cos (2 π x^{3}) d x

Apply u-substitution

: \int \frac{u cos ( 2 π u ^{\frac{3}{2}} )}{2 x} d u

= 2 \cdot \int \frac{cos ( 2 π u )}{3} d u

Take the constant out:

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

= 2 \cdot \frac{1}{3} \cdot \int cos (2 π u) d u

Apply u-substitution

: \int cos (v) \frac{1}{2 π} d v

= 2 \cdot \frac{1}{3} \cdot \int cos (v) \frac{1}{2 π} d v

Take the constant out:

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

= 2 \cdot \frac{1}{3} \cdot \frac{1}{2 π} \cdot \int cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= 2 \cdot \frac{1}{3} \cdot \frac{1}{2 π} sin (v)

Substitute back

= 2 \cdot \frac{1}{3} \cdot \frac{1}{2 π} sin (2 x^{3} π)

Simplify

2 \cdot \frac{1}{3} \cdot \frac{1}{2 π} sin (2 x^{3} π) : \frac{1}{3 π} sin (2 π x^{3})

= \frac{1}{3 π} sin (2 π x^{3})

Add a constant to the solution

= \frac{1}{3 π} sin (2 π x^{3}) + C

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What is the integral of 2x^2cos(2pix^3) ?
The integral of 2x^2cos(2pix^3) is 1/(3pi)sin(2pix^3)+C