Let I=∫(5x+3)√(2x 1) dx Let 5x+3=λ(2x 1)+μ Comparing the coefficients, we get ⇒ 2λ=5⇒ λ=52 ⇒ λ+μ=3⇒ 52+μ=3 ⇒ μ=3+52=6+52=112 =∫(λ ...

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Functions

∫ 5x+3 √ 2x-1...

Question

$\int (5 x + 3) \sqrt{2 x - 1} d x$

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Solution

Let $I = \int (5 x + 3) \sqrt{2 x - 1} d x$

Let $5 x + 3 = λ (2 x - 1) + μ$
Comparing the coefficients, we get

$\Rightarrow 2 λ = 5 \Rightarrow λ = \frac{5}{2}$
$\Rightarrow - λ + μ = 3 \Rightarrow - \frac{5}{2} + μ = 3$
$\Rightarrow μ = 3 + \frac{5}{2} = \frac{6 + 5}{2} = \frac{11}{2}$
$= \int (λ (2 x - 1) + μ) \sqrt{2 x - 1} d x$
$= \int (\frac{5}{2} (2 x - 1) + \frac{11}{2}) \sqrt{2 x - 1} d x$

$∴ I = \int (5 x + 3) \sqrt{2 x - 1} d x$
$= \frac{5}{2} \int {(2 x - 1)}^{1 + \frac{1}{2}} d x + \frac{11}{2} \int {(2 x - 1)}^{\frac{1}{2}} d x$
$= \frac{5}{2} \int {(2 x - 1)}^{\frac{3}{2}} d x + \frac{11}{2} \int {(2 x - 1)}^{\frac{1}{2}} d x$

$= \frac{5}{2} \frac{{(2 x - 1)}^{\frac{3}{2} + 1}}{2 (\frac{3}{2} + 1)} + \frac{11}{2} \frac{{(2 x - 1)}^{\frac{1}{2} + 1}}{2 (\frac{1}{2} + 1)} + c$
$= \frac{5}{2} \frac{{(2 x - 1)}^{\frac{5}{2}}}{2 (\frac{5}{2})} + \frac{11}{2} \frac{{(2 x - 1)}^{\frac{3}{2}}}{2 (\frac{3}{2})} + c$
$= \frac{5}{2} \frac{{(2 x - 1)}^{\frac{5}{2}}}{5} + \frac{11}{2} \frac{{(2 x - 1)}^{\frac{3}{2}}}{3} + c$
$= \frac{1}{2} {(2 x - 1)}^{\frac{5}{2}} + \frac{11}{6} {(2 x - 1)}^{\frac{3}{2}} + c$

$= \frac{1}{6} {(2 x - 1)}^{\frac{3}{2}} [3 (2 x - 1) + 11] + c$
$= \frac{1}{6} {(2 x - 1)}^{\frac{3}{2}} [6 x - 3 + 11] + c$
$= \frac{1}{6} {(2 x - 1)}^{\frac{3}{2}} [6 x + 8] + c$
$= \frac{1}{6} {(2 x - 1)}^{\frac{3}{2}} [2 (3 x + 4)] + c$
$= \frac{2}{6} {(2 x - 1)}^{\frac{3}{2}} (3 x + 4) + c$
$= \frac{1}{3} {(2 x - 1)}^{\frac{3}{2}} (3 x + 4) + c$

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