Question

x - 4/x²+2x-15 dx

Integrate​

Answer 1

Hello there !

∫x - 4/x²+2x-15. dx

x²/2 +4 /x + 2x²/2 -15x

= x² /2 + x² + 4/x -15x

= 3x²/2 + 4/x -15x

Hope it helps you.

Answer 2

Given Integrand,

$\displaystyle \sf f(x) = \int \dfrac{x - 4}{ {x}^{2} + 2x - 15} dx$

Adding and subtracting one in denominator,

$\longrightarrow \sf \displaystyle \sf \: f(x) = \int \dfrac{x - 4}{( {x}^{2} + 2x + 1) - 16 } dx \\ \\ \longrightarrow \sf \displaystyle \sf \: f(x) = \int \dfrac{x - 4}{( {x} + 1) {}^{2} - {4}^{2} }dx \\ \\ \longrightarrow \sf \displaystyle \sf \: f(x) = \int \dfrac{x - 4}{(x + 1 - 4)(x + 1 + 4) }dx \\ \\ \longrightarrow \sf \displaystyle \sf \: f(x) = \int \dfrac{x - 4}{(x - 3)(x + 5) }dx$

Now,

$\sf \: \dfrac{x - 4}{(x - 3)(x + 5) } = \dfrac{a}{x + 5} + \dfrac{b}{x - 3} \\ \\ \implies \sf \: x - 4 = a(x - 3) + b(x + 5)$

When x = 3, b = -1/8

When x = -5, a = 8/9

Therefore,

$\longrightarrow \sf \displaystyle \sf \: f(x) = \dfrac{8}{9} \int \dfrac{dx}{x + 5} - \dfrac{1}{8} \int \dfrac{dx}{x - 3} \\ \\ \longrightarrow \sf \displaystyle \boxed{\boxed{\sf \: f(x) = \dfrac{8}{9} ln(x + 5) - \dfrac{1}{8} ln(x - 3) + C}}$