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

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

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integral of cos(x^2)x^3

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

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

Solution steps

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

\int \frac{x ^{2} + 3 x + 2}{x ^{3} + 4 x} d x

\int - 5 x d x

\int \frac{e ^{2 x}}{7 - e ^{4 x}} d x

\int \frac{4 x ^{3} - 8 x ^{2} + 4 x + 3}{4 x ^{2} - 8 x + 5} d x

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