Evaluate the integral: I = integral (sqrt(x(x+1)))/(sqrt
Evaluate the integral: $$ I = \int \frac{\sqrt{x(x+1)}}{\sqrt x+\sqrt{x+1}}\,dx. $$
Explanation of Method
Rationalizing by the conjugate $\sqrt{x+1}-\sqrt x$ collapses the denominator to $1$, leaving $\sqrt x\sqrt{x+1}(\sqrt{x+1}-\sqrt x)$, which expands into two elementary power-rule integrals.
- Rationalize by the conjugate.
- Expand the resulting integrand.
- Integrate each piece.
Step 1 · Rationalize
Multiplying by $\dfrac{\sqrt{x+1}-\sqrt x}{\sqrt{x+1}-\sqrt x}$, the denominator becomes $(x+1)-x=1$:
$$ I = \int\sqrt x\sqrt{x+1}\left(\sqrt{x+1}-\sqrt x\right)dx = \int x^{1/2}(x+1)\,dx - \int x(x+1)^{1/2}\,dx. $$
Step 2 · First piece
$$ \int x^{1/2}(x+1)\,dx = \int\left(x^{3/2}+x^{1/2}\right)dx = \frac{2x^{5/2}}5 + \frac{2x^{3/2}}3 + C_1. $$
Step 3 · Second piece
Let $u=x+1$, so $x=u-1$:
$$ \int x(x+1)^{1/2}\,dx = \int(u-1)u^{1/2}\,du = \frac{2u^{5/2}}5 - \frac{2u^{3/2}}3 + C_2 = \frac{2(x+1)^{5/2}}5 - \frac{2(x+1)^{3/2}}3 + C_2. $$
Step 4 · Combine
$$ I = \frac{2x^{5/2}}5 + \frac{2x^{3/2}}3 - \frac{2(x+1)^{5/2}}5 + \frac{2(x+1)^{3/2}}3 + C. $$
Summary Table
| Piece | Antiderivative |
|---|---|
| $x^{3/2}+x^{1/2}$ | $\tfrac25x^{5/2}+\tfrac23x^{3/2}$ |
| $(u-1)u^{1/2}$ | $\tfrac25u^{5/2}-\tfrac23u^{3/2}$, $u=x+1$ |
Additional Notes
- The domain is $x\geq0$.
- The result was verified by differentiation at $x=1$ and $x=2$: the derivative matches the integrand $\sqrt{x(x+1)}/(\sqrt x+\sqrt{x+1})$ exactly at both points.
Result
The solution to the integral is: