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What is antiderivative in sin x+ln x?

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rouche | Student, Undergraduate | eNoter

Posted June 22, 2013 at 2:23 PM via web

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What is antiderivative in sin x+ln x?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 22, 2013 at 3:00 PM (Answer #1)

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You need to evaluate the anti-derivative of the given function, hence, you need to find the primitive integral of function such that:

`int (sin x + ln x) dx`

Using the property of linearity of integral you need to split the original integral into two simpler integrals, such that:

`int (sin x + ln x) dx = int sin x dx + int ln x dx`

`int (sin x + ln x) dx = -cos x + int ln x dx`

You need to use integration by parts to evaluate `int ln x dx` such that:

`int udv = uv - int vdu`

Considering `u = ln x` and `dv = dx` yields:

`int ln x dx = x*ln x - int x*(1/x)dx`

`int ln x dx = x*ln x - int dx`

`int ln x dx = x*ln x - x + c`

Factoring out ` x` yields:

`int ln x dx = x*(ln x - 1) + c`

Hence, evaluating the anti-derivative of the given function, yields `int (sin x + ln x) dx = -cos x + x*(ln x - 1) + c` .

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