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

The integral of 4x^2*sin(5x^3) is -4/15 cos(5x^3)+C

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

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

- \frac{4}{15} cos (5 x^{3}) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{sin ( u )}{15} d u

= 4 \cdot \int \frac{sin ( u )}{15} d u

Take the constant out:

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

= 4 \cdot \frac{1}{15} \cdot \int sin (u) d u

Use the common integral:

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

= 4 \cdot \frac{1}{15} (- cos (u))

Substitute back

u = 5 x^{3}

= 4 \cdot \frac{1}{15} (- cos (5 x^{3}))

Simplify

4 \cdot \frac{1}{15} (- cos (5 x^{3})) : - \frac{4}{15} cos (5 x^{3})

= - \frac{4}{15} cos (5 x^{3})

Add a constant to the solution

= - \frac{4}{15} cos (5 x^{3}) + C

f (z) = e^{\frac{z}{( z - 5 )}}

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What is the integral of 4x^2*sin(5x^3) ?
The integral of 4x^2*sin(5x^3) is -4/15 cos(5x^3)+C