What is the integral of x^3cos(3x^4+2) ?

The integral of x^3cos(3x^4+2) is 1/24 sin(6x^4)+C

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integral of x^3cos(3x^4+2)

\int x^{3} cos (3 x^{4} + 2) d x

\frac{1}{24} sin (6 x^{4}) + C

Solution steps

\int x^{3} cos (3 x^{4} + 2) d x

Rewrite using trig identities

= \int x^{3} (cos^{2} (3 x^{4}) - sin^{2} (3 x^{4})) d x

Simplify

x^{3} (cos^{2} (3 x^{4}) - sin^{2} (3 x^{4})) : x^{3} cos (6 x^{4})

= \int x^{3} cos (6 x^{4}) d x

Apply u-substitution

= \int \frac{cos ( u )}{24} d u

Take the constant out:

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

= \frac{1}{24} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{24} sin (u)

Substitute back

u = 6 x^{4}

= \frac{1}{24} sin (6 x^{4})

Add a constant to the solution

= \frac{1}{24} sin (6 x^{4}) + C

What is the integral of x^3cos(3x^4+2) ?
The integral of x^3cos(3x^4+2) is 1/24 sin(6x^4)+C