What is the integral of x^5(cos(x^3)) ?

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

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integral of x^5(cos(x^3))

\int x^{5} (cos (x^{3})) d x

\frac{1}{3} (x^{3} sin (x^{3}) + cos (x^{3})) + C

Solution steps

\int x^{5} cos (x^{3}) d x

Apply u-substitution

: \int \frac{u cos ( u )}{3} d u

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

Take the constant out:

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

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

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= \frac{1}{3} (u sin (u) - \int sin (u) d u)

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

= \frac{1}{3} (u sin (u) - (- cos (u)))

Substitute back

u = x^{3}

= \frac{1}{3} (x^{3} sin (x^{3}) - (- cos (x^{3})))

Simplify

= \frac{1}{3} (x^{3} sin (x^{3}) + cos (x^{3}))

Add a constant to the solution

= \frac{1}{3} (x^{3} sin (x^{3}) + cos (x^{3})) + C

\frac{d}{d z} (ln (\frac{a ^{2} - z ^{2}}{a ^{2} + z ^{2}}))

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\int_{0}^{7} \frac{1}{9 + x ^{2}} d x

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\int \frac{x ^{2}}{( x ^{2} + 6 )} d x

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