What is the integral of 1/3*x^3cos(x^4+2) ?

The integral of 1/3*x^3cos(x^4+2) is 1/12 sin(x^4+2)+C

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integral of 1/3*x^3cos(x^4+2)

\int \frac{1}{3} \cdot x^{3} cos (x^{4} + 2) d x

\frac{1}{12} sin (x^{4} + 2) + C

Solution steps

\int \frac{1}{3} x^{3} cos (x^{4} + 2) d x

Take the constant out:

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

= \frac{1}{3} \cdot \int x^{3} cos (x^{4} + 2) d x

Apply u-substitution

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

= \frac{1}{3} \cdot \int \frac{cos ( u )}{4} d u

Take the constant out:

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

= \frac{1}{3} \cdot \frac{1}{4} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{3} \cdot \frac{1}{4} sin (u)

Substitute back

u = x^{4} + 2

= \frac{1}{3} \cdot \frac{1}{4} sin (x^{4} + 2)

Simplify

\frac{1}{3} \cdot \frac{1}{4} sin (x^{4} + 2) : \frac{1}{12} sin (x^{4} + 2)

= \frac{1}{12} sin (x^{4} + 2)

Add a constant to the solution

= \frac{1}{12} sin (x^{4} + 2) + C

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What is the integral of 1/3*x^3cos(x^4+2) ?
The integral of 1/3*x^3cos(x^4+2) is 1/12 sin(x^4+2)+C