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

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

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

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

x^{2} sin (x^{2}) + cos (x^{2}) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 2 \cdot \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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = x^{2}

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

Simplify

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

= x^{2} sin (x^{2}) + cos (x^{2})

Add a constant to the solution

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

\int 8 cos (6 x) d x

x \to 0 lim (\frac{x ^{2}}{x ^{2}})

\frac{\partial}{\partial y} (x^{2} - y)

\int e^{x^{2} + 1} 2 x d x

\frac{d y}{d x} (\frac{1}{x ^{8}}) - 15 y = 12

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