What is the integral of x^2sqrt(x^3+33) ?

The integral of x^2sqrt(x^3+33) is 2/9 (x^3+33)^{3/2}+C

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integral of x^2sqrt(x^3+33)

\int x^{2} x^{3} + 33 d x

\frac{2}{9} (x^{3} + 33)^{\frac{3}{2}} + C

Solution steps

\int x^{2} x^{3} + 33 d x

Apply u-substitution

: \int \frac{u}{3} d u

= \int \frac{u}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = x^{3} + 33

= \frac{1}{3} \cdot \frac{2}{3} (x^{3} + 33)^{\frac{3}{2}}

Simplify

\frac{1}{3} \cdot \frac{2}{3} (x^{3} + 33)^{\frac{3}{2}} : \frac{2}{9} (x^{3} + 33)^{\frac{3}{2}}

= \frac{2}{9} (x^{3} + 33)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{9} (x^{3} + 33)^{\frac{3}{2}} + C

\frac{d y}{d x} + y^{5} x + 2 y = 0

\int \frac{1}{x ^{5} x ^{2} - 1} d x

x^{2} - 3 x

\frac{d}{d x} (- \frac{( a x + b - d ) ( x ^{2} - 1 )}{x ^{2}})

\frac{d}{d x} (arccsc (\frac{x}{2}))

What is the integral of x^2sqrt(x^3+33) ?
The integral of x^2sqrt(x^3+33) is 2/9 (x^3+33)^{3/2}+C