What is the integral of (-2x)/(\sqrt[4]{x+2)} ?

The integral of (-2x)/(\sqrt[4]{x+2)} is -2(4/7 (x+2)^{7/4}-8/3 (x+2)^{3/4})+C

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integral of (-2x)/(\sqrt[4]{x+2)}

- 2 (\frac{4}{7} (x + 2)^{\frac{7}{4}} - \frac{8}{3} (x + 2)^{\frac{3}{4}}) + C

Solution steps

Take the constant out:

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

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

= - 2 (\frac{4}{7} u^{\frac{7}{4}} - \frac{8}{3} u^{\frac{3}{4}})

Substitute back

u = x + 2

= - 2 (\frac{4}{7} (x + 2)^{\frac{7}{4}} - \frac{8}{3} (x + 2)^{\frac{3}{4}})

Add a constant to the solution

= - 2 (\frac{4}{7} (x + 2)^{\frac{7}{4}} - \frac{8}{3} (x + 2)^{\frac{3}{4}}) + C

x \to 0 lim (x^{2} + 3 x - 1)

\frac{d}{d x} (4 x^{2} - 5)

\frac{d ^{2}}{d x ^{2}} (3 x - \frac{7}{x})

\frac{d}{d x} (ln (\frac{3 x}{x + 2}))

\frac{d}{d x} (32 cos (2 x))

What is the integral of (-2x)/(\sqrt[4]{x+2)} ?
The integral of (-2x)/(\sqrt[4]{x+2)} is -2(4/7 (x+2)^{7/4}-8/3 (x+2)^{3/4})+C