Question

integrate (11x - 2x ^ 2)/(2x + 2) dx​

Answer 1

Step-by-step explanation:

(11x-2x^2)/(2x+2)dx

(11x_2=9x^4x)dx

36x^dx

Answer 2

$\displaystyle \sf \int \: \frac{11x - {2x}^{2} }{2x + 2} \: dx$

$\displaystyle \sf\int \: \frac{11x}{2x + 2} - \frac{ {2x}^{2} }{2x + 2} \: dx$

$\displaystyle \sf\int \: \frac{11x}{2x + 2} - \frac{ {2x}^{2} }{2(x + 1)} \: dx$

Cancel 2 from the fraction.

$\displaystyle \sf \int \: \frac{11x}{2x + 2} - \frac{ {x}^{2} }{x + 1} \: dx$

$\displaystyle \sf\int \: \frac{11x}{2x + 2} \: dx - \int \: \frac{ {x}^{2} }{ x + 1} \: dx$

Evaluate by substitution.

$\displaystyle \sf \int \frac{11t-22}{4t} \: dt - \int \frac{t² - 2t+1}{t} \: dt$

$\displaystyle \sf \int \frac{11(t -2) }{4t} \: dt - \int \frac{t²}{t} - \frac{2t}{t} + \frac{1}{t} \: dt$

$\displaystyle \sf \frac{11}{4} × \int \frac{t-2}{t} \: dt - \int \: t - 2 + \frac{1}{t} \: dt$

Divide , Integrate , Substitute back t = 2x + 1 and t = x + 1.

$\displaystyle \sf \frac{11}{2} x + \frac{11}{2} - \frac{11}{2} \times ln( |2x +2| ) - \frac{ {x}^{2} - 2x - 3}{2} - ln( |x + 1| )$

$\displaystyle \sf \green{\: \frac{13x + 14 - {x}^{2} }{2} - \frac{11}{2} \times ln( |2x + 2| ) - ln( |x + 1| ) + c}$

Additional information:

$\begin{gathered}\boxed{\begin{array}{c|c}\bf f(x)&\bf\displaystyle\int\rm f(x)\:dx\\ \\ \frac{\qquad\qquad}{}&\frac{\qquad\qquad}{}\\ \rm k&\rm kx+C\\ \\ \rm sin(x)&\rm-cos(x)+C\\ \\ \rm cos(x)&\rm sin(x)+C\\ \\ \rm{sec}^{2}(x)&\rm tan(x)+C\\ \\ \rm{cosec}^{2}(x)&\rm-cot(x)+C\\ \\ \rm sec(x)\ tan(x)&\rm sec(x)+C\\ \\ \rm cosec(x)\ cot(x)&\rm-cosec(x)+C\\ \\ \rm tan(x)&\rm log(sec(x))+C\\ \\ \rm\dfrac{1}{x}&\rm log(x)+C\\ \\ \rm{e}^{x}&\rm{e}^{x}+C\\ \\ \rm x^{n},n\neq-1&\rm\dfrac{x^{n+1}}{n+1}+C\end{array}}\end{gathered}$