What is the integral of x^3sin(4x^4-pi) ?

The integral of x^3sin(4x^4-pi) is 1/16 cos(4x^4)+C

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integral of x^3sin(4x^4-pi)

\int x^{3} sin (4 x^{4} - π) d x

\frac{1}{16} cos (4 x^{4}) + C

Solution steps

\int x^{3} sin (4 x^{4} - π) d x

Rewrite using trig identities

= \int - x^{3} sin (4 x^{4}) d x

Take the constant out:

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

= - \int x^{3} sin (4 x^{4}) d x

Apply Integration By Parts

: \frac{1}{4} x^{4} sin (4 x^{4}) - \int 4 x^{7} cos (4 x^{4}) d x

= - (\frac{1}{4} x^{4} sin (4 x^{4}) - \int 4 x^{7} cos (4 x^{4}) d x)

\int 4 x^{7} cos (4 x^{4}) d x = \frac{1}{16} (4 x^{4} sin (4 x^{4}) + cos (4 x^{4}))

= - (\frac{1}{4} x^{4} sin (4 x^{4}) - \frac{1}{16} (4 x^{4} sin (4 x^{4}) + cos (4 x^{4})))

Simplify

- (\frac{1}{4} x^{4} sin (4 x^{4}) - \frac{1}{16} (4 x^{4} sin (4 x^{4}) + cos (4 x^{4}))) : \frac{1}{16} cos (4 x^{4})

= \frac{1}{16} cos (4 x^{4})

Add a constant to the solution

= \frac{1}{16} cos (4 x^{4}) + C

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What is the integral of x^3sin(4x^4-pi) ?
The integral of x^3sin(4x^4-pi) is 1/16 cos(4x^4)+C