What is the integral of 2x^3sin(5x^4) ?

The integral of 2x^3sin(5x^4) is -1/10 cos(5x^4)+C

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integral of 2x^3sin(5x^4)

\int 2 x^{3} sin (5 x^{4}) d x

- \frac{1}{10} cos (5 x^{4}) + C

Solution steps

\int 2 x^{3} sin (5 x^{4}) d x

Take the constant out:

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

= 2 \cdot \int x^{3} sin (5 x^{4}) d x

Apply u-substitution

: \int \frac{sin ( u )}{20} d u

= 2 \cdot \int \frac{sin ( u )}{20} d u

Take the constant out:

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

= 2 \cdot \frac{1}{20} \cdot \int sin (u) d u

Use the common integral:

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

= 2 \cdot \frac{1}{20} (- cos (u))

Substitute back

u = 5 x^{4}

= 2 \cdot \frac{1}{20} (- cos (5 x^{4}))

Simplify

2 \cdot \frac{1}{20} (- cos (5 x^{4})) : - \frac{1}{10} cos (5 x^{4})

= - \frac{1}{10} cos (5 x^{4})

Add a constant to the solution

= - \frac{1}{10} cos (5 x^{4}) + C

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What is the integral of 2x^3sin(5x^4) ?
The integral of 2x^3sin(5x^4) is -1/10 cos(5x^4)+C