Question

Make a substitution to express the integrand as a rational function and then evaluate the integral. int_(25)^(81) sqrt(x)/(x-1) dx

Answer 1

Let y = √x, so that y ² = x and 2y dy = dx. Then the integral becomes

`\displaystyle \int_(25)^(81) (\sqrt x)/(x-1)\,\mathrm dx = \int_(√(25))^(√(81)) \frac y{y^2-1}(2y\,\mathrm dy) = 2 \int_5^9 (y^2)/(y^2-1)\,\mathrm dy`

Now,

y ² / (y ² - 1) = 1 + 1 / (y ² - 1) = 1 + 1/2 (1/(y - 1) - 1/(y + 1))

so integrating gives us

`\displaystyle 2\int_5^9(y^2)/(y^2-1)\,\mathrm dy= \int_5^9\left(2+\frac1{y-1}-\frac1{y+1}\right)\,\mathrm dy \\\\= (2y+\ln|y-1|-\ln|y+1|)\bigg|_5^9 \\\\= \boxed{8+\ln\left(\frac65\right)}`