What is the integral of x\sqrt[4]{6x+5} ?

The integral of x\sqrt[4]{6x+5} is 1/9 (1/9 (6x+5)^{9/4}-(6x+5)^{5/4})+C

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integral of x\sqrt[4]{6x+5}

\frac{1}{9} (\frac{1}{9} (6 x + 5)^{\frac{9}{4}} - (6 x + 5)^{\frac{5}{4}}) + C

Solution steps

Apply u-substitution

: \int \frac{u ^{4} ( u ^{4} - 5 )}{9} d u

= \int \frac{u ^{4} ( u ^{4} - 5 )}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int u^{4} (u^{4} - 5) d u

Expand

u^{4} (u^{4} - 5) : u^{8} - 5 u^{4}

= \frac{1}{9} \cdot \int u^{8} - 5 u^{4} 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}{9} (\int u^{8} d u - \int 5 u^{4} d u)

\int u^{8} d u = \frac{u ^{9}}{9}

\int 5 u^{4} d u = u^{5}

= \frac{1}{9} (\frac{u ^{9}}{9} - u^{5})

Substitute back

Simplify

= \frac{1}{9} (\frac{1}{9} (6 x + 5)^{\frac{9}{4}} - (6 x + 5)^{\frac{5}{4}})

Add a constant to the solution

= \frac{1}{9} (\frac{1}{9} (6 x + 5)^{\frac{9}{4}} - (6 x + 5)^{\frac{5}{4}}) + C

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What is the integral of x\sqrt[4]{6x+5} ?
The integral of x\sqrt[4]{6x+5} is 1/9 (1/9 (6x+5)^{9/4}-(6x+5)^{5/4})+C