Question

explain integrals of some particular functions​

Answer 1

Answer:

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Answer 2

$\huge {\boxed {\mathbb {QUESTION}}}$

explain integration by partial functions

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$A\: rational \:function\: is\: a\: ratio\: of\: two\\ polynomials\:\frac{ P(x)} {Q(x)} , where \:Q(x)\\ \cancel {=} 0.$

$Now, \:if \:the\: degree\: of\: P(x) \:is\: lesser\: than\: the\\ degree\: of\: Q(x), \:then\: it \:is\: a \:proper\: fraction,\\ else\: it \:is\: an \:improper\: fraction.$

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$The\: integrand\: is\: a \:proper\: rational\: function\\ by \:using\: the\: form\: of \:partial\: fraction$

$\frac{1} {[(x + 1) (x + 2)]} =\frac{ A} {(x + 1)} + \frac{B} {(x + 2)} ---- (1)$

$Solving\: this\: equation$

$A (x + 2) + B (x + 1) = 1$

$Ax + 2A + Bx + B = 1$

$x (A + B) + (2A + B) = 1$

$LHS = RHS,$

$A + B = 0\: and\: 2A + B = 1$

$solving \:these\: two\: equations$

$A = 1\: and\: B = – 1.$

$\therefore$

$\implies \frac{1} {[(x + 1) (x + 2)]} = \frac{1} {(x + 1)} –\frac{ 1} {(x + 2)}$

$\implies \int \frac{dx} {[(x + 1) (x + 2)]} = \int {dx} {(x + 1)} – \int{ dx} {(x + 2)}$

$\implies log |x + 1| – log |x + 2| + C$

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