What is the integral of 1/(x^{1/3)+x^{1/4}} ?

The integral of 1/(x^{1/3)+x^{1/4}} is

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integral of 1/(x^{1/3)+x^{1/4}}

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

12 (\frac{1}{8} x^{\frac{2}{3}} - \frac{1}{7} x^{\frac{7}{12}} + \frac{1}{6} x^{\frac{1}{2}} - \frac{1}{5} x^{\frac{5}{12}} + \frac{1}{4} x^{\frac{1}{3}} - \frac{1}{3} x^{\frac{1}{4}} + \frac{1}{2} x^{\frac{1}{6}} - x^{\frac{1}{12}} + ln x^{\frac{1}{12}} + 1) + C

Solution steps

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

Apply u-substitution

: \int \frac{12 u ^{8}}{u + 1} d u

= \int \frac{12 u ^{8}}{u + 1} d u

Take the constant out:

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

= 12 \cdot \int \frac{u ^{8}}{u + 1} d u

Long division

\frac{u ^{8}}{u + 1} : u^{7} - u^{6} + u^{5} - u^{4} + u^{3} - u^{2} + u - 1 + \frac{1}{u + 1}

= 12 \cdot \int u^{7} - u^{6} + u^{5} - u^{4} + u^{3} - u^{2} + u - 1 + \frac{1}{u + 1} 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

= 12 (\int u^{7} d u - \int u^{6} d u + \int u^{5} d u - \int u^{4} d u + \int u^{3} d u - \int u^{2} d u + \int u d u - \int 1 d u + \int \frac{1}{u + 1} d u)

\int u^{7} d u = \frac{u ^{8}}{8}

\int u^{6} d u = \frac{u ^{7}}{7}

\int u^{5} d u = \frac{u ^{6}}{6}

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

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

\int u^{2} d u = \frac{u ^{3}}{3}

\int u d u = \frac{u ^{2}}{2}

\int 1 d u = u

\int \frac{1}{u + 1} d u = ln ∣ u + 1 ∣

= 12 (\frac{u ^{8}}{8} - \frac{u ^{7}}{7} + \frac{u ^{6}}{6} - \frac{u ^{5}}{5} + \frac{u ^{4}}{4} - \frac{u ^{3}}{3} + \frac{u ^{2}}{2} - u + ln ∣ u + 1 ∣)

Substitute back

u = x^{\frac{1}{12}}

= 12 \frac{( x ^{\frac{1}{12}} ) ^{8}}{8} - \frac{( x ^{\frac{1}{12}} ) ^{7}}{7} + \frac{( x ^{\frac{1}{12}} ) ^{6}}{6} - \frac{( x ^{\frac{1}{12}} ) ^{5}}{5} + \frac{( x ^{\frac{1}{12}} ) ^{4}}{4} - \frac{( x ^{\frac{1}{12}} ) ^{3}}{3} + \frac{( x ^{\frac{1}{12}} ) ^{2}}{2} - x^{\frac{1}{12}} + ln x^{\frac{1}{12}} + 1

Simplify

12 \frac{( x ^{\frac{1}{12}} ) ^{8}}{8} - \frac{( x ^{\frac{1}{12}} ) ^{7}}{7} + \frac{( x ^{\frac{1}{12}} ) ^{6}}{6} - \frac{( x ^{\frac{1}{12}} ) ^{5}}{5} + \frac{( x ^{\frac{1}{12}} ) ^{4}}{4} - \frac{( x ^{\frac{1}{12}} ) ^{3}}{3} + \frac{( x ^{\frac{1}{12}} ) ^{2}}{2} - x^{\frac{1}{12}} + ln x^{\frac{1}{12}} + 1 : 12 (\frac{1}{8} x^{\frac{2}{3}} - \frac{1}{7} x^{\frac{7}{12}} + \frac{1}{6} x^{\frac{1}{2}} - \frac{1}{5} x^{\frac{5}{12}} + \frac{1}{4} x^{\frac{1}{3}} - \frac{1}{3} x^{\frac{1}{4}} + \frac{1}{2} x^{\frac{1}{6}} - x^{\frac{1}{12}} + ln x^{\frac{1}{12}} + 1)

= 12 (\frac{1}{8} x^{\frac{2}{3}} - \frac{1}{7} x^{\frac{7}{12}} + \frac{1}{6} x^{\frac{1}{2}} - \frac{1}{5} x^{\frac{5}{12}} + \frac{1}{4} x^{\frac{1}{3}} - \frac{1}{3} x^{\frac{1}{4}} + \frac{1}{2} x^{\frac{1}{6}} - x^{\frac{1}{12}} + ln x^{\frac{1}{12}} + 1)

Add a constant to the solution

= 12 (\frac{1}{8} x^{\frac{2}{3}} - \frac{1}{7} x^{\frac{7}{12}} + \frac{1}{6} x^{\frac{1}{2}} - \frac{1}{5} x^{\frac{5}{12}} + \frac{1}{4} x^{\frac{1}{3}} - \frac{1}{3} x^{\frac{1}{4}} + \frac{1}{2} x^{\frac{1}{6}} - x^{\frac{1}{12}} + ln x^{\frac{1}{12}} + 1) + C

75.6 x + \frac{36590.4}{x ^{2}}

\int \frac{1}{x ^{3} - x ^{2}} d x

\int e^{c o s (29 t)} sin (29 t) d t

What is the integral of 1/(x^{1/3)+x^{1/4}} ?
The integral of 1/(x^{1/3)+x^{1/4}} is