What is the integral of 6xcos(x^2) ?

The integral of 6xcos(x^2) is 3sin(x^2)+C

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integral of 6xcos(x^2)

\int 6 x cos (x^{2}) d x

3 sin (x^{2}) + C

Solution steps

\int 6 x cos (x^{2}) d x

Take the constant out:

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

= 6 \cdot \int x cos (x^{2}) d x

Apply u-substitution

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

= 6 \cdot \int \frac{cos ( u )}{2} d u

Take the constant out:

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

= 6 \cdot \frac{1}{2} \cdot \int cos (u) d u

Use the common integral:

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

= 6 \cdot \frac{1}{2} sin (u)

Substitute back

u = x^{2}

= 6 \cdot \frac{1}{2} sin (x^{2})

Simplify

6 \cdot \frac{1}{2} sin (x^{2}) : 3 sin (x^{2})

= 3 sin (x^{2})

Add a constant to the solution

= 3 sin (x^{2}) + C

\frac{\partial}{\partial x} (x^{8} + 5 x y^{9})

\frac{\partial}{\partial y} (x e^{x^{2} + y^{2}})

\frac{d y}{d x} + x^{2} y = 1

x \to 0 lim (8^{- x})

3 ln (x) - \frac{x ^{2}}{24}