(\integral) (x – 10C1 x2 + 10C2x3 –10C3x4 .....+ 10C10x11) d... | MATH - INTEGRATION BY SUBSTITUTION | SolveeIt

$\int$ (x – ¹⁰C₁ x² + ¹⁰C₂x³ –¹⁰C₃x⁴ .....+ ¹⁰C₁₀x¹¹) dx equals:

$\frac{(1 - x)^{11}}{11}$ – $\frac{(1 - x)^{10}}{10}$ + c

$\frac{(1 - x)^{12}}{12}$ – $\frac{(1 - x)^{11}}{11}$ + c

$\frac{(1 - x)^{10}}{10}$ – $\frac{(1 - x)^{11}}{11}$ + c

$\frac{(1 - x)^{11}}{11}$ – $\frac{(1 - x)^{12}}{12}$ + c

Answer

$\frac{(1 - x)^{12}}{12}$ – $\frac{(1 - x)^{11}}{11}$ + c

Explanation

Given expression

= x (1–¹⁰C₁ x + ¹⁰C₂x² + .....+ ¹⁰C₁₀x¹⁰)

= x(1– x)¹⁰

\ I = $\int_{}^{}{x(1 - x)^{10}}$ dx

Let I –x = t – dx = dt

\ I = – $\int_{}^{}{(1 - t)t^{10}}$ dt

= $\frac { - \mathrm { t } ^ { 11 } } { 11 }$ + $\frac{t^{12}}{12}$ + c

Question: \(\int\) (x – <sup>10</sup>C<sub>1</sub> x<sup>2</sup> + <sup>10</sup>C<sub>2</sub>x<sup>3</sup> –<s...