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How can I resolve this indefinite integral?

$$ \int \left({8,387x+1 \over 9,41x+1} + \sin(9,326x + 1)\right) dx $$

I'm blocked here

$$ \int\left({{8387 \over 1000}x+1}\over{{941\over100}x+1} \right)dx + \int\left(\sin\left({9326\over1000}x+1\right) \right)dx $$

please anyone can help me? thanks

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Replace those strange numbers with some friendly ones so you do not get distracted while doing and learning the substitution method. Then apply what you learned to the general case. –  Maesumi Jan 17 '13 at 13:47
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2 Answers

up vote 4 down vote accepted

As said above, I also think you are getting distracted by the "strange" numbers.

Assume you have

$\displaystyle\int\dfrac{ax+1}{bx+1}dx+\displaystyle\int\sin(cx+1)dx$,

where $a,b,c$ are some constants, then you can write

$\displaystyle\int\dfrac{ax}{bx+1}dx+\displaystyle\int\dfrac{1}{bx+1}dx+\displaystyle\int\sin(cx+1)dx$,

now just proceed by rewriting the first term as $\dfrac{a}{b}-\dfrac{a/b}{bx+1}$, this is, now you have

$\displaystyle\int\left( \dfrac{a}{b}-\dfrac{a/b}{bx+1} \right)dx+\displaystyle\int\dfrac{1}{bx+1}dx+\displaystyle\int\sin(cx+1)dx$,

and now you are able to integrate everything by declaring new variables as already suggested: $u=bx+1$ and $v=cx+1$.

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Do a substitution $w = 9.41 \, x + 1$ for the first and $y = 9.236 \, x + 1$ for the second integral.

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but how can i resolve the 8,387x+1? –  Sam Jan 17 '13 at 12:39
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