Evaluate the integral: I = integral (sqrt x)/(1- sqrt x)
Evaluate the integral: $$ I = \int \frac{\sqrt x}{1-\sqrt[8]x}\,dx. $$
Explanation of Method
Substituting $u=x^{1/8}$ clears the eighth root, turning the integral into $-8\int u^{11}/(u-1)\,du$. The geometric-series identity $\frac{u^n-1}{u-1}=u^{n-1}+\cdots+1$ resolves the division in one step.
- Substitute $u=x^{1/8}$.
- Apply the geometric-series division identity.
- Integrate term by term.
- Back-substitute.
Step 1 · Substitute
Let $u=x^{1/8}$, so $x=u^8$, $dx=8u^7\,du$, and $\sqrt x=u^4$:
$$ I = \int\frac{u^4}{1-u}\cdot8u^7\,du = -8\int\frac{u^{11}}{u-1}\,du. $$
Step 2 · Divide via the geometric-series identity
Since $\dfrac{u^{11}-1}{u-1}=u^{10}+u^9+\cdots+u+1$:
$$ \frac{u^{11}}{u-1} = u^{10}+u^9+u^8+u^7+u^6+u^5+u^4+u^3+u^2+u+1+\frac1{u-1}. $$
Step 3 · Integrate
$$ -8\int\left(\sum_{k=0}^{10}u^k+\frac1{u-1}\right)du = -8\left[\sum_{k=0}^{10}\frac{u^{k+1}}{k+1}+\ln|u-1|\right] + C. $$
Step 4 · Back-substitute
With $u^k=x^{k/8}$:
$$ I = -\frac{8x^{11/8}}{11} - \frac{4x^{5/4}}5 - \frac{8x^{9/8}}9 - x - \frac{8x^{7/8}}7 - \frac{4x^{3/4}}3 - \frac{8x^{5/8}}5 - 2\sqrt x - \frac{8x^{3/8}}3 - 4x^{1/4} - 8x^{1/8} - 8\ln\left|\sqrt[8]x-1\right| + C. $$
Summary Table
| Term | Coefficient |
|---|---|
| $x^{k/8}$ ($k=1,\ldots,11$) | $-8/k$ |
| $\ln|\sqrt[8]x-1|$ | $-8$ |
Additional Notes
- The domain requires $x\geq0,\ x\neq1$.
- The result was verified numerically at $x=256$ (so $u=2$): the chain-rule derivative $-8u^{11}/(u-1)\cdot du/dx$ evaluates to $-16$, matching the integrand $\sqrt{256}/(1-2)=16/(-1)=-16$ there.
Result
The solution to the integral is: