What is the integral of x^3cos(1+x^2) ?

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

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integral of x^3cos(1+x^2)

\int x^{3} cos (1 + x^{2}) d x

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

Solution steps

\int x^{3} cos (1 + x^{2}) d x

Apply u-substitution

: \int \frac{( u - 1 ) cos ( u )}{2} d u

= \int \frac{( u - 1 ) 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 - 1) cos (u) d u

Apply Integration By Parts

: sin (u) (u - 1) - \int sin (u) d u

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

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

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

Substitute back

u = 1 + x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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