Integral príklady - substitučná metodá, per-partes a elementárne zlomky
Odpovědi
Diskuze
subst:
y=√( ln(x) )
y^2=ln(x)
dy=1/2 * 1/√( ln(x) ) * 1/x * dx
∫ √( ln(x) ) /( x* ln(x)) * dx =
∫ y /( x* ln(x)) * dx =
dy=1/2 * 1/y * 1/x * dx
2*y*dy=1/x * dx
∫ y /( x* ln(x)) * dx =
∫ y /y^2 * 2*y*dy =
∫ 2*dy = 2y + C =
=2√( ln(x) ) + C
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http://www.wolframalpha.com/input/?i=%E2%88%AB+%E2%88%9A(+ln(x)+)+%2F(+x*+ln(x))+*+dx
b) ∫√x * ln(x) * dx
per partes
u'=√x; u=x^(3/2) / (3/2) = 2/3 * x^(3/2)
v=ln(x); v'=1/x
∫u'*v = u*v - ∫u*v'
∫√x * ln(x) * dx =
2/3 * x^(3/2) * ln(x) - 2/3 * ∫x^(3/2) / x * dx =
2/3 * x^(3/2) * ln(x) - 2/3 * ∫x^(1/2) * dx =
2/3 * x^(3/2) * ln(x) - 2/3 * x^(3/2) / (3/2) + C =
= 2/3 * [ x^(3/2) * ln(x) - 2/3 * x^(3/2) ] + C
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http://www.wolframalpha.com/input/?i=%E2%88%AB%E2%88%9Ax+*+ln(x)+*+dx+%3D
c) ∫(3x + 4)/(x^2 + 49)*dx
(x^2 + 49)'=2x
∫(3x + 4)/(x^2 + 49)dx =
(arctan(x/a))'= a/(x^2 + a^2)
1/a * (arctan(x/a))'= 1/(x^2 + a^2)
3/2*∫2x/(x^2 + 49)*dx + 4 * 1/(x^2 + 7^2)*dx =
= 3/2* ln|x^2 + 49| + 4/7 * atctg/(x/7) + C
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http://www.wolframalpha.com/input/?i=%E2%88%AB(3x+%2B+4)%2F(x%5E2+%2B+49)*dx