What is the integral of x^3sqrt(x^2+26) ?

The integral of x^3sqrt(x^2+26) is 1/5 (x^2+26)^{5/2}-26/3 (x^2+26)^{3/2}+C

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integral of x^3sqrt(x^2+26)

\int x^{3} x^{2} + 26 d x

\frac{1}{5} (x^{2} + 26)^{\frac{5}{2}} - \frac{26}{3} (x^{2} + 26)^{\frac{3}{2}} + C

Solution steps

\int x^{3} x^{2} + 26 d x

Apply u-substitution

: \int (u^{2} - 26) u^{2} d u

= \int (u^{2} - 26) u^{2} d u

Expand

(u^{2} - 26) u^{2} : u^{4} - 26 u^{2}

= \int u^{4} - 26 u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{4} d u - \int 26 u^{2} d u

\int u^{4} d u = \frac{u ^{5}}{5}

\int 26 u^{2} d u = \frac{26 u ^{3}}{3}

= \frac{u ^{5}}{5} - \frac{26 u ^{3}}{3}

Substitute back

u = x^{2} + 26

= \frac{( x ^{2} + 26 ) ^{5}}{5} - \frac{26 ( x ^{2} + 26 ) ^{3}}{3}

Simplify

\frac{( x ^{2} + 26 ) ^{5}}{5} - \frac{26 ( x ^{2} + 26 ) ^{3}}{3} : \frac{1}{5} (x^{2} + 26)^{\frac{5}{2}} - \frac{26}{3} (x^{2} + 26)^{\frac{3}{2}}

= \frac{1}{5} (x^{2} + 26)^{\frac{5}{2}} - \frac{26}{3} (x^{2} + 26)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{5} (x^{2} + 26)^{\frac{5}{2}} - \frac{26}{3} (x^{2} + 26)^{\frac{3}{2}} + C

\frac{1}{( s ( s ^{2} + 0.4 s + 4 ) )}

\int_{1}^{x} 1 + 9 x d x

\frac{\partial}{\partial x} (1 + x ln (x y - 7))

\frac{\partial ^{2}}{\partial x \partial y} ((1 + 2 x y^{2})^{3})

n = 1 \sum \infty 4 (2.1)^{n - 1}

What is the integral of x^3sqrt(x^2+26) ?
The integral of x^3sqrt(x^2+26) is 1/5 (x^2+26)^{5/2}-26/3 (x^2+26)^{3/2}+C