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

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

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

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

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

Solution steps

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

Apply u-substitution

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

= 2 \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

= 2 \cdot \frac{1}{4} \cdot \int cos (u) d u

Use the common integral:

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

= 2 \cdot \frac{1}{4} sin (u)

Substitute back

u = x^{4}

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

Simplify

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

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

Add a constant to the solution

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

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