First slide

Methods of integration

Question

$Evaluate (x - 5) \sqrt{x^{2} + x} d x = \frac{{(x^{2} + x)}^{\frac{3}{2}}}{3} - A (x + \frac{1}{2}) \sqrt{{(x + \frac{1}{2})}^{2} - {(\frac{1}{2})}^{2}} + B \log |(x + \frac{1}{2}) + \sqrt{{(x + \frac{1}{2})}^{2} - {(\frac{1}{2})}^{2}}| + C$

Moderate

A

A= $\frac{- 11}{4}, B = \frac{- 11}{8}$
B

$A = \frac{- 11}{4}, B = \frac{11}{8}$
C

$A = \frac{11}{4}, B = \frac{- 11}{8}$
D

$A = \frac{11}{4}, B = \frac{11}{8}$

Solution

$Let (x - 5) = λ \frac{d}{d x} (x^{2} + x) + μ . Then, x - 5 = λ (2 x + 1) + μ$
$Comparing coefficients of like powers of x, we get 1 = 2 λ and λ + μ = - 5 or λ = \frac{1}{2} and μ = - \frac{11}{2}, Therefore,$
$\begin{array}{l} \int (x - 5) \sqrt{x^{2} + x} d x \\ = \int [\frac{1}{2} (2 x + 1) - \frac{11}{2}] \sqrt{x^{2} + x} d x \\ = \frac{1}{2} \int (2 x + 1) \sqrt{x^{2} + x} d x - \frac{11}{2} \int \sqrt{x^{2} + x} d x \\ = \frac{1}{2} \int \sqrt{t} d t - \frac{11}{2} \int \sqrt{{(x + \frac{1}{2})}^{2} - {(\frac{1}{2})}^{2}} d x w h e r e t = x^{2} + x \\ = \frac{1}{2} \frac{t^{3 / 2}}{3 / 2} - \frac{11}{2} [\frac{1}{2} (x + \frac{1}{2}) \sqrt{{(x + \frac{1}{2})}^{2} - {(\frac{1}{2})}^{2}} + \frac{1}{2} {(\frac{1}{2})}^{2} \log |(x + \frac{1}{2}) + \sqrt{{(x + \frac{1}{2})}^{2} - {(\frac{1}{2})}^{2}}|] + C \\ = \frac{{(x^{2} + x)}^{\frac{3}{2}}}{3} - \frac{11}{4} (x + \frac{1}{2}) \sqrt{{(x + \frac{1}{2})}^{2} - {(\frac{1}{2})}^{2}} - \frac{11}{8} \log |(x + \frac{1}{2}) + \sqrt{{(x + \frac{1}{2})}^{2} - {(\frac{1}{2})}^{2}}| + C \end{array}$

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