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Solving integrals/ (3/x)-2*sin(5*x)+1

Integral of (3/x)-2*sin(5*x)+1 dx

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The solution

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∫01((−2sin⁡(5x)+3x)+1) dx\int\limits_{0}^{1} \left(\left(- 2 \sin{\left(5 x \right)} + \frac{3}{x}\right) + 1\right)\, dx 
Integral(3/x - 2*sin(5*x) + 1, (x, 0, 1))
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:

        ∫(−2sin⁡(5x)) dx=−2∫sin⁡(5x) dx\int \left(- 2 \sin{\left(5 x \right)}\right)\, dx = - 2 \int \sin{\left(5 x \right)}\, dx

        1. Let u=5xu = 5 x.

          Then let du=5dxdu = 5 dx and substitute du5\frac{du}{5}:

          ∫sin⁡(u)5 du\int \frac{\sin{\left(u \right)}}{5}\, du

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

            ∫sin⁡(u) du=∫sin⁡(u) du5\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}

            1. The integral of sine is negative cosine:

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

            So, the result is: −cos⁡(u)5- \frac{\cos{\left(u \right)}}{5}

          Now substitute uu back in:

          −cos⁡(5x)5- \frac{\cos{\left(5 x \right)}}{5}

        So, the result is: 2cos⁡(5x)5\frac{2 \cos{\left(5 x \right)}}{5}

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

        ∫3x dx=3∫1x dx\int \frac{3}{x}\, dx = 3 \int \frac{1}{x}\, dx

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

        So, the result is: 3log⁡(x)3 \log{\left(x \right)}

      The result is: 3log⁡(x)+2cos⁡(5x)53 \log{\left(x \right)} + \frac{2 \cos{\left(5 x \right)}}{5}

    1. The integral of a constant is the constant times the variable of integration:

      ∫1 dx=x\int 1\, dx = x

    The result is: x+3log⁡(x)+2cos⁡(5x)5x + 3 \log{\left(x \right)} + \frac{2 \cos{\left(5 x \right)}}{5}

  2. Add the constant of integration:

    x+3log⁡(x)+2cos⁡(5x)5+constantx + 3 \log{\left(x \right)} + \frac{2 \cos{\left(5 x \right)}}{5}+ \mathrm{constant}

The answer is:

x+3log⁡(x)+2cos⁡(5x)5+constantx + 3 \log{\left(x \right)} + \frac{2 \cos{\left(5 x \right)}}{5}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                       
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 | /3                 \                         2*cos(5*x)
 | |- - 2*sin(5*x) + 1| dx = C + x + 3*log(x) + ----------
 | \x                 /                             5     
 |                                                        
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∫((−2sin⁡(5x)+3x)+1) dx=C+x+3log⁡(x)+2cos⁡(5x)5\int \left(\left(- 2 \sin{\left(5 x \right)} + \frac{3}{x}\right) + 1\right)\, dx = C + x + 3 \log{\left(x \right)} + \frac{2 \cos{\left(5 x \right)}}{5}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-5000050000
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The answer [src] 
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Numerical answer [src] 
132.984803276164
132.984803276164

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

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