What is the integral of ((x+2)^2)/(\sqrt[3]{3-x)} ?

The integral of ((x+2)^2)/(\sqrt[3]{3-x)} is -(x+2)^2(3-x)^{2/3}+1/8 (5x^2+2x-151)(3-x)^{2/3}+C

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integral of ((x+2)^2)/(\sqrt[3]{3-x)}

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

- (x + 2)^{2} (3 - x)^{\frac{2}{3}} + \frac{1}{8} (5 x^{2} + 2 x - 151) (3 - x)^{\frac{2}{3}} + C

Solution steps

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

Apply u-substitution

: \int - \frac{( - u + 5 ) ^{2}}{3 u} d u

= \int - \frac{( - u + 5 ) ^{2}}{3 u} d u

Take the constant out:

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

= - \int \frac{( - u + 5 ) ^{2}}{3 u} d u

Apply Integration By Parts

: u^{\frac{2}{3}} (- u + 5)^{2} - \int \frac{6 3 u u ^{\frac{5}{3}} - 30 3 u u ^{\frac{2}{3}} - u ^{2} + 10 u - 25}{3 u ^{\frac{1}{3}}} d u

= - (u^{\frac{2}{3}} (- u + 5)^{2} - \int \frac{6 3 u u ^{\frac{5}{3}} - 30 3 u u ^{\frac{2}{3}} - u ^{2} + 10 u - 25}{3 u ^{\frac{1}{3}}} d u)

\int \frac{6 3 u u ^{\frac{5}{3}} - 30 3 u u ^{\frac{2}{3}} - u ^{2} + 10 u - 25}{3 u ^{\frac{1}{3}}} d u = \frac{1}{8} u^{\frac{2}{3}} (5 u^{2} - 32 u - 100)

= - (u^{\frac{2}{3}} (- u + 5)^{2} - \frac{1}{8} u^{\frac{2}{3}} (5 u^{2} - 32 u - 100))

Substitute back

u = 3 - x

= - ((3 - x)^{\frac{2}{3}} (- (3 - x) + 5)^{2} - \frac{1}{8} (3 - x)^{\frac{2}{3}} (5 (3 - x)^{2} - 32 (3 - x) - 100))

Simplify

- ((3 - x)^{\frac{2}{3}} (- (3 - x) + 5)^{2} - \frac{1}{8} (3 - x)^{\frac{2}{3}} (5 (3 - x)^{2} - 32 (3 - x) - 100)) : - (x + 2)^{2} (3 - x)^{\frac{2}{3}} + \frac{1}{8} (5 x^{2} + 2 x - 151) (3 - x)^{\frac{2}{3}}

= - (x + 2)^{2} (3 - x)^{\frac{2}{3}} + \frac{1}{8} (5 x^{2} + 2 x - 151) (3 - x)^{\frac{2}{3}}

Add a constant to the solution

= - (x + 2)^{2} (3 - x)^{\frac{2}{3}} + \frac{1}{8} (5 x^{2} + 2 x - 151) (3 - x)^{\frac{2}{3}} + C

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

n = 1 \sum \infty \frac{9 n}{7 n + 1}

y = x, y = 0, y = x - 2

x \to 3 + lim (\frac{2}{x - 3})

y^{^{'}} = y - 4 y^{2}

What is the integral of ((x+2)^2)/(\sqrt[3]{3-x)} ?
The integral of ((x+2)^2)/(\sqrt[3]{3-x)} is -(x+2)^2(3-x)^{2/3}+1/8 (5x^2+2x-151)(3-x)^{2/3}+C