For x

Question

(D) Given $I = \int \frac{x-x^2}{\sqrt{1-x}} d x$. Since $x-x^2 = x(1-x)$,we have $I = \int \frac{x(1-x)}{\sqrt{1-x}} d x = \int x \sqrt{1-x} d x$. Le

Vedclass · Accepted Answer

$-\frac{2}{15}(1-x)^{3 / 2}(3x+2)+c$

Vedclass · Answer

(D) Given $I = \int \frac{x-x^2}{\sqrt{1-x}} d x$. Since $x-x^2 = x(1-x)$,we have $I = \int \frac{x(1-x)}{\sqrt{1-x}} d x = \int x \sqrt{1-x} d x$. Let $1-x = t^2$,then $dx = -2t dt$. Substituting these into the integral: $I = \int (1-t^2) t (-2t) dt = 2 \int (t^4 - t^2) dt$. Integrating with respect to $t$: $I = 2 \left[ \frac{t^5}{5} - \frac{t^3}{3} \right] + c = \frac{2}{15} t^3 (3t^2 - 5) + c$. Substituting $t = \sqrt{1-x}$ back: $I = \frac{2}{15} (1-x)^{3/2} [3(1-x) - 5] + c = \frac{2}{15} (1-x)^{3/2} (-3x - 2) + c = -\frac{2}{15} (1-x)^{3/2} (3x+2) + c$.

For $x < 1$,evaluate $\int \frac{x-x^2}{\sqrt{1-x}} d x$.

Similar Questions

$\int \frac{e^{2025+x} - e^{2025-x}}{e^{2026+x} + e^{2026-x}} dx = $ . . . . . . + $C$

Find the following integral:
$\int \sin ^{3} x \cos ^{2} x \, dx$

$\int \frac{d x}{(x+2) \sqrt{x+1}} = $

$\int \sin^{\frac{-1}{2}}x \cos^{\frac{-7}{2}}x \, dx = $

$\int \frac{\log (x + \sqrt {1 + x^2})}{\sqrt {1 + x^2}} \, dx = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module