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Solving integrals/ sin(y)+y*sin(x)+1/x

Integral of sin(y)+y*sin(x)+1/x dx

Limits of integration:

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Piecewise:

The solution

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

  1. Integrate term-by-term:

    1. Integrate term-by-term:

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

        ∫ysin⁡(x) dx=y∫sin⁡(x) dx\int y \sin{\left(x \right)}\, dx = y \int \sin{\left(x \right)}\, dx

        1. The integral of sine is negative cosine:

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

        So, the result is: −ycos⁡(x)- y \cos{\left(x \right)}

      1. The integral of sine is negative cosine:

        ∫sin⁡(y) dx=−cos⁡(y)\int \sin{\left(y \right)}\, dx = - \cos{\left(y \right)}

      The result is: −ycos⁡(x)−cos⁡(y)- y \cos{\left(x \right)} - \cos{\left(y \right)}

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

    The result is: −ycos⁡(x)+log⁡(x)−cos⁡(y)- y \cos{\left(x \right)} + \log{\left(x \right)} - \cos{\left(y \right)}

  2. Add the constant of integration:

    −ycos⁡(x)+log⁡(x)−cos⁡(y)+constant- y \cos{\left(x \right)} + \log{\left(x \right)} - \cos{\left(y \right)}+ \mathrm{constant}

The answer is:

−ycos⁡(x)+log⁡(x)−cos⁡(y)+constant- y \cos{\left(x \right)} + \log{\left(x \right)} - \cos{\left(y \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
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∫((ysin⁡(x)+sin⁡(y))+1x) dx=C−ycos⁡(x)+log⁡(x)−cos⁡(y)\int \left(\left(y \sin{\left(x \right)} + \sin{\left(y \right)}\right) + \frac{1}{x}\right)\, dx = C - y \cos{\left(x \right)} + \log{\left(x \right)} - \cos{\left(y \right)}
Simplify
The answer [src] 
oo + x*sin(y) - y*cos(x) + log(x)
xsin⁡(y)−ycos⁡(x)+log⁡(x)+∞x \sin{\left(y \right)} - y \cos{\left(x \right)} + \log{\left(x \right)} + \infty 
=
= 
oo + x*sin(y) - y*cos(x) + log(x)
xsin⁡(y)−ycos⁡(x)+log⁡(x)+∞x \sin{\left(y \right)} - y \cos{\left(x \right)} + \log{\left(x \right)} + \infty 
oo + x*sin(y) - y*cos(x) + log(x)

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

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