What is the integral of y\sqrt[3]{y+5} ?

The integral of y\sqrt[3]{y+5} is 3/7 (y+5)^{7/3}-15/4 (y+5)^{4/3}+C

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integral of y\sqrt[3]{y+5}

\frac{3}{7} (y + 5)^{\frac{7}{3}} - \frac{15}{4} (y + 5)^{\frac{4}{3}} + C

Solution steps

Apply u-substitution

Expand

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^{\frac{4}{3}} d u = \frac{3}{7} u^{\frac{7}{3}}

= \frac{3}{7} u^{\frac{7}{3}} - \frac{15}{4} u^{\frac{4}{3}}

Substitute back

u = y + 5

= \frac{3}{7} (y + 5)^{\frac{7}{3}} - \frac{15}{4} (y + 5)^{\frac{4}{3}}

Add a constant to the solution

= \frac{3}{7} (y + 5)^{\frac{7}{3}} - \frac{15}{4} (y + 5)^{\frac{4}{3}} + C

x \to + 0 lim (\frac{2 ^{x} - 3 ^{x}}{x})

e^{9 e^{x}}

\frac{d y}{d x} = \frac{1}{6 x - 42}

y = \frac{e ^{4 x^{2} + 1}}{8}

2 x + 5 x

What is the integral of y\sqrt[3]{y+5} ?
The integral of y\sqrt[3]{y+5} is 3/7 (y+5)^{7/3}-15/4 (y+5)^{4/3}+C