integral calculus logarithmic functions
Pano po naging sagot sa problem na integral of dx all over x(1+x^2) e 1/2 ln x^2 over 1+x^2 +c
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Pano po naging sagot sa problem na integral of dx all over x(1+x^2) e 1/2 ln x^2 over 1+x^2 +c
Re: integral calculus logarithmic functions
hindi po malinaw kung ano talaga ang equation. You can use symbols of grouping (brackets) para mas madali maintindihan.
Re: integral calculus logarithmic functions
In reply to Re: integral calculus logarithmic functions by Jhun Vert
patulong nga po. 3x^4+4x^3+16x^2+20x+9 over (x+20)(x^2+3)^2 pasagot nga po yan hirap po kase
Re: integral calculus logarithmic functions
In reply to Re: integral calculus logarithmic functions by Jeanvill Palad…
Ano ang tanong dito? Mahirap manghula kuna ano ang ibig mong sabihin.
Re: integral calculus logarithmic functions
Binasa ko uli ang post mo at naintindihan ko na. You ask why
$\displaystyle \int \dfrac{dx}{x(1 + x^2)} = \dfrac{1}{2} \, \ln \left( \dfrac{x^2}{1 + x^2} \right) + C$
am I right? If so, here is the detail:
$\dfrac{1}{x(1 + x^2)} = \dfrac{A}{x} + \dfrac{Bx + C}{1 + x^2}$
$1 = A(1 + x^2) + Bx^2 + Cx$
When x = 0, A = 1
Equate x2: 0 = A + B, B = -1
Equate x: C = 0
Thus,
$\displaystyle \int \dfrac{dx}{x(1 + x^2)} = \int \left( \dfrac{1}{x} - \dfrac{x}{1 + x^2} \right) \, dx$
$\displaystyle \int \dfrac{dx}{x(1 + x^2)} = \int \dfrac{dd}{x} - \dfrac{1}{2} \int \dfrac{2x \, dx}{1 + x^2}$
$\displaystyle \int \dfrac{dx}{x(1 + x^2)} = \ln x - \dfrac{1}{2} \ln (1 + x^2) + C$
$\displaystyle \int \dfrac{dx}{x(1 + x^2)} = \dfrac{1}{2} \left[ 2\ln x - \ln (1 + x^2) \right] + C$
$\displaystyle \int \dfrac{dx}{x(1 + x^2)} = \dfrac{1}{2} \left[ \ln x^2 - \ln (1 + x^2) \right] + C$
$\displaystyle \int \dfrac{dx}{x(1 + x^2)} = \dfrac{1}{2} \, \ln \left( \dfrac{x^2}{1 + x^2} \right) + C$
as we expect it to be.