What is the integral of ((sqrt(x)+1))/(\sqrt[4]{x)} ?

The integral of ((sqrt(x)+1))/(\sqrt[4]{x)} is 4/5 x^{5/4}+4/3 x^{3/4}+C

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integral of ((sqrt(x)+1))/(\sqrt[4]{x)}

\int \frac{( x + 1 )}{4 x} d x

\frac{4}{5} x^{\frac{5}{4}} + \frac{4}{3} x^{\frac{3}{4}} + C

Solution steps

\int \frac{x + 1}{4 x} d x

Expand

\frac{x + 1}{4 x} : 4 x + \frac{1}{4 x}

= \int 4 x + \frac{1}{4 x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 4 x d x + \int \frac{1}{4 x} d x

\int 4 x d x = \frac{4}{5} x^{\frac{5}{4}}

\int \frac{1}{4 x} d x = \frac{4}{3} x^{\frac{3}{4}}

= \frac{4}{5} x^{\frac{5}{4}} + \frac{4}{3} x^{\frac{3}{4}}

Add a constant to the solution

= \frac{4}{5} x^{\frac{5}{4}} + \frac{4}{3} x^{\frac{3}{4}} + C

\int - \frac{3}{2} x d x

\int \frac{10}{( 4 x ^{3} - 4 x ^{2} + 8 x )} d x

\frac{d}{d x} (∣ sin (6 x) ∣)

\frac{d}{d x} ((2 x - 1) 3 - 4 x)

s^{2}

What is the integral of ((sqrt(x)+1))/(\sqrt[4]{x)} ?
The integral of ((sqrt(x)+1))/(\sqrt[4]{x)} is 4/5 x^{5/4}+4/3 x^{3/4}+C