What is the integral of x(x^2+3)^5 ?

The integral of x(x^2+3)^5 is 1/12 (x^2+3)^6+C

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integral of x(x^2+3)^5

\int x (x^{2} + 3)^{5} d x

\frac{1}{12} (x^{2} + 3)^{6} + C

Solution steps

\int x (x^{2} + 3)^{5} d x

Apply u-substitution

: \int \frac{u ^{5}}{2} d u

= \int \frac{u ^{5}}{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^{5} d u

Apply the Power Rule

: \frac{u ^{6}}{6}

= \frac{1}{2} \cdot \frac{u ^{6}}{6}

Substitute back

u = x^{2} + 3

= \frac{1}{2} \cdot \frac{( x ^{2} + 3 ) ^{6}}{6}

Simplify

\frac{1}{2} \cdot \frac{( x ^{2} + 3 ) ^{6}}{6} : \frac{1}{12} (x^{2} + 3)^{6}

= \frac{1}{12} (x^{2} + 3)^{6}

Add a constant to the solution

= \frac{1}{12} (x^{2} + 3)^{6} + C

\frac{d}{d x} (3 x^{5} - 4 x^{4} + 9 x - 6)

\int \frac{sin ( x )}{2} d x

\frac{d}{d x} (sin (y) - y sin (x))

\int sec^{5} (2 x) tan (2 x) d x

x \to 1 + lim (\frac{6}{x - 1})

What is the integral of x(x^2+3)^5 ?
The integral of x(x^2+3)^5 is 1/12 (x^2+3)^6+C