What is the integral of x\sqrt[3]{(x+3)^5} ?

The integral of x\sqrt[3]{(x+3)^5} is 3/88 (8x-9)(x+3)^{8/3}+C

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integral of x\sqrt[3]{(x+3)^5}

\frac{3}{88} (8 x - 9) (x + 3)^{\frac{8}{3}} + C

Solution steps

Apply u-substitution

Take the constant out:

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

Expand

= \frac{1}{5} \cdot \int \frac{1}{u ^{\frac{4}{15}}} - \frac{3}{u ^{\frac{7}{15}}} 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

= \frac{1}{5} (\int \frac{1}{u ^{\frac{4}{15}}} d u - \int \frac{3}{u ^{\frac{7}{15}}} d u)

\int \frac{1}{u ^{\frac{4}{15}}} d u = \frac{15}{11} u^{\frac{11}{15}}

\int \frac{3}{u ^{\frac{7}{15}}} d u = \frac{45}{8} u^{\frac{8}{15}}

= \frac{1}{5} (\frac{15}{11} u^{\frac{11}{15}} - \frac{45}{8} u^{\frac{8}{15}})

Substitute back

u = (x + 3)^{5}

= \frac{1}{5} (\frac{15}{11} ((x + 3)^{5})^{\frac{11}{15}} - \frac{45}{8} ((x + 3)^{5})^{\frac{8}{15}})

Simplify

\frac{1}{5} (\frac{15}{11} ((x + 3)^{5})^{\frac{11}{15}} - \frac{45}{8} ((x + 3)^{5})^{\frac{8}{15}}) : \frac{3}{88} (8 x - 9) (x + 3)^{\frac{8}{3}}

= \frac{3}{88} (8 x - 9) (x + 3)^{\frac{8}{3}}

Add a constant to the solution

= \frac{3}{88} (8 x - 9) (x + 3)^{\frac{8}{3}} + C

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x \to + 0 lim (\frac{2 sin ( 2 x )}{x})

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10 y^{^{''}} - 7 y^{^{'}} - 12 y = 0

What is the integral of x\sqrt[3]{(x+3)^5} ?
The integral of x\sqrt[3]{(x+3)^5} is 3/88 (8x-9)(x+3)^{8/3}+C