What is the integral of x(x^2+3)^{12} ?

The integral of x(x^2+3)^{12} is 1/26 (x^2+3)^{13}+C

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integral of x(x^2+3)^{12}

\int x (x^{2} + 3)^{12} d x

\frac{1}{26} (x^{2} + 3)^{13} + C

Solution steps

\int x (x^{2} + 3)^{12} d x

Apply u-substitution

: \int \frac{u ^{12}}{2} d u

= \int \frac{u ^{12}}{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^{12} d u

Apply the Power Rule

: \frac{u ^{13}}{13}

= \frac{1}{2} \cdot \frac{u ^{13}}{13}

Substitute back

u = x^{2} + 3

= \frac{1}{2} \cdot \frac{( x ^{2} + 3 ) ^{13}}{13}

Simplify

\frac{1}{2} \cdot \frac{( x ^{2} + 3 ) ^{13}}{13} : \frac{1}{26} (x^{2} + 3)^{13}

= \frac{1}{26} (x^{2} + 3)^{13}

Add a constant to the solution

= \frac{1}{26} (x^{2} + 3)^{13} + C

\frac{\partial}{\partial y} (x^{2} - 3 x y)

30 y^{^{''}} - 25 y^{^{'}} = 0

\int x e^{x} d x

\int_{0}^{1} \frac{e ^{z} + 1}{e ^{z} + z} d z

\frac{d y}{d x} = 40 - 3 y

What is the integral of x(x^2+3)^{12} ?
The integral of x(x^2+3)^{12} is 1/26 (x^2+3)^{13}+C