If In- xna - x1- 2 dx and I2-7x- a-x- 3-2-4a-7I1 evaluate - - -

I_n=∫ x^n(a x)^1/ 2 dx #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; I #160; #160; ...

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Byju's Answer

Standard XII

Mathematics

Integration by Partial Fractions

If In=∫ xna...

Question

If $I_{n} = \int x^{n} (a - x)^{1 / 2} d x$ and $I_{2} = \frac{α}{7} x^{β} (a - x^{γ})^{3 / 2} + \frac{4 a}{7} I_{1}$ evaluate $α + β + γ$

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Solution

$I_{n} = \int x^{n} (a - x)^{1 / 2} d x$
$I I$
$= - \frac{2}{3} x^{n} (a - x)^{3 / 2} + \frac{2 n}{3} \int x^{n - 1} (a - x)^{3 / 2} d x$
$I_{n} = - \frac{2}{3} x^{n} (a - x)^{3 / 2} + \frac{2 n}{3} \int x^{n - 1} (a - x) (a - x)^{1 / 2} d x$
$I_{n} = - \frac{2}{3} x^{n} (a - x)^{3 / 2} + \frac{2 n a}{3}$
$= \int x^{n - 1} (a - x)^{1 / 2} - \frac{2 n}{3} \int x^{n} (a - x)^{1 / 2} d x$
$\Rightarrow [1 + \frac{2 n}{3}] I_{n} = - \frac{2}{3} x^{n} (a - x)^{3 / 2} + \frac{2 a n}{3} I_{n - 1}$
$(2 n + 3) I_{n} = - 2 x^{n} (a - x)^{3 / 2} + 2 a n I_{n - 1}$
$7 I_{2} = - 2 x^{2} (a - x)^{3 / 2} + 4 a I_{1}$
$I_{2} = - \frac{2}{7} x^{2} (a - x)^{3 / 2} + \frac{4 a}{7} I_{1} \Rightarrow α = 2, β = 2, γ = 1$

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If In=∫xn(a−x)1/2dx and I2=α7xβ(a−xγ)3/2+4a7I1 evaluate α+β+γ

If $I_{n} = \int x^{n} (a - x)^{1 / 2} d x$ and $I_{2} = \frac{α}{7} x^{β} (a - x^{γ})^{3 / 2} + \frac{4 a}{7} I_{1}$ evaluate $α + β + γ$