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Solving integrals/ 1/x-sin10x/x

Integral of 1/x-sin10x/x dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1                   
  /                   
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 |  /1   sin(10*x)\   
 |  |- - ---------| dx
 |  \x       x    /   
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0
∫01(−sin⁡(10x)x+1x) dx\int\limits_{0}^{1} \left(- \frac{\sin{\left(10 x \right)}}{x} + \frac{1}{x}\right)\, dx 
Integral(1/x - sin(10*x)/x, (x, 0, 1))
Detail solution

  1. Integrate term-by-term:

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫(−sin⁡(10x)x) dx=−∫sin⁡(10x)x dx\int \left(- \frac{\sin{\left(10 x \right)}}{x}\right)\, dx = - \int \frac{\sin{\left(10 x \right)}}{x}\, dx

        SiRule(a=10, b=0, context=sin(10*x)/x, symbol=x)

      So, the result is: −Si⁡(10x)- \operatorname{Si}{\left(10 x \right)}

    1. The integral of 1x\frac{1}{x} is log⁡(x)\log{\left(x \right)}.

    The result is: log⁡(x)−Si⁡(10x)\log{\left(x \right)} - \operatorname{Si}{\left(10 x \right)}

  2. Add the constant of integration:

    log⁡(x)−Si⁡(10x)+constant\log{\left(x \right)} - \operatorname{Si}{\left(10 x \right)}+ \mathrm{constant}

The answer is:

log⁡(x)−Si⁡(10x)+constant\log{\left(x \right)} - \operatorname{Si}{\left(10 x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                          
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 | /1   sin(10*x)\                           
 | |- - ---------| dx = C - Si(10*x) + log(x)
 | \x       x    /                           
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∫(−sin⁡(10x)x+1x) dx=C+log⁡(x)−Si⁡(10x)\int \left(- \frac{\sin{\left(10 x \right)}}{x} + \frac{1}{x}\right)\, dx = C + \log{\left(x \right)} - \operatorname{Si}{\left(10 x \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-500010000
Plot the graph f(x)
The answer [src] 
oo - Si(10)
−Si⁡(10)+∞- \operatorname{Si}{\left(10 \right)} + \infty 
=
= 
oo - Si(10)
−Si⁡(10)+∞- \operatorname{Si}{\left(10 \right)} + \infty 
oo - Si(10)
Numerical answer [src] 
42.432098539774
42.432098539774

    Use the examples entering the upper and lower limits of integration.

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