Animated Solution for Mathematics - Indefinite Integration: Find the indefinite integral $\int \left(\frac{1}{\sqrt[3]{x}+\sqrt[4]{x}}+\frac{\ln (1+\sqrt[6]{x})}{\sqrt[3]{x}+\sqrt{x}}\right)dx$

To solve the integral

we decompose the expression into two manageable parts: $I_1$ and $I_2$. By addressing these separately, we simplify the complexity of the fractional exponents.

We define $I_1=\int \frac{dx}{x^{1/3}+x^{1/4}}$. The least common multiple of the denominators $3$ and $4$ is $12$.

We apply the substitution $x=y^{12}$, which implies $dx=12y^{11}dy$. Substituting these into the integral yields:

Performing polynomial long division on $\frac{y^8}{y+1}$, we obtain the alternating series:

Integrating term by term and substituting $y=x^{1/12}$ back, we find:

We define $I_2=\int \frac{\ln (1+x^{1/6})}{x^{1/3}+x^{1/2}}dx$. The LCM of the denominators $6,3,$ and $2$ is $6$.

We substitute $x=z^6$, which gives $dx=6z^{5}dz$. This transforms the integral into:

To simplify further, we use the substitution $t=1+z$, implying $z=t-1$ and $dz=dt$:

Expanding the integral, we have:

Using integration by parts for the first term and simple substitution for the second, we get:

Substituting $t=1+x^{1/6}$ back into the expression, the final result is the sum of $I_1$ and $I_2$ plus the constant of integration $C$.