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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{u cos ( u )}{2} d u

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = x^{2}

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

Simplify

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

Add a constant to the solution

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

\frac{d}{d x} ((4 x^{3} - 7 x)^{7})

\int x 2 x - x^{2} d x

\int (\frac{8 + 5 x}{1 + x ^{2}}) d x

\frac{d}{d x} (4 x^{4} + x^{3} - \frac{9 x ^{2}}{2} + 8 x)

sin (π x), \frac{1}{2}

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