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Solving integrals/ cos(3/2x-1)dx

Integral of cos(3/2x-1)dx dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1                
  /                
 |                 
 |     /3*x    \   
 |  cos|--- - 1| dx
 |     \ 2     /   
 |                 
/                  
0
∫01cos⁡(3x2−1) dx\int\limits_{0}^{1} \cos{\left(\frac{3 x}{2} - 1 \right)}\, dx 
Integral(cos(3*x/2 - 1), (x, 0, 1))
Detail solution

  1. Let u=3x2−1u = \frac{3 x}{2} - 1.

    Then let du=3dx2du = \frac{3 dx}{2} and substitute 2du3\frac{2 du}{3}:

    ∫2cos⁡(u)3 du\int \frac{2 \cos{\left(u \right)}}{3}\, du

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

      ∫cos⁡(u) du=2∫cos⁡(u) du3\int \cos{\left(u \right)}\, du = \frac{2 \int \cos{\left(u \right)}\, du}{3}

      1. The integral of cosine is sine:

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

      So, the result is: 2sin⁡(u)3\frac{2 \sin{\left(u \right)}}{3}

    Now substitute uu back in:

    2sin⁡(3x2−1)3\frac{2 \sin{\left(\frac{3 x}{2} - 1 \right)}}{3}

  2. Now simplify:

    2sin⁡(3x2−1)3\frac{2 \sin{\left(\frac{3 x}{2} - 1 \right)}}{3}

  3. Add the constant of integration:

    2sin⁡(3x2−1)3+constant\frac{2 \sin{\left(\frac{3 x}{2} - 1 \right)}}{3}+ \mathrm{constant}

The answer is:

2sin⁡(3x2−1)3+constant\frac{2 \sin{\left(\frac{3 x}{2} - 1 \right)}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                           /3*x    \
 |                       2*sin|--- - 1|
 |    /3*x    \               \ 2     /
 | cos|--- - 1| dx = C + --------------
 |    \ 2     /                3       
 |                                     
/
∫cos⁡(3x2−1) dx=C+2sin⁡(3x2−1)3\int \cos{\left(\frac{3 x}{2} - 1 \right)}\, dx = C + \frac{2 \sin{\left(\frac{3 x}{2} - 1 \right)}}{3}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-2
Plot the graph f(x)
The answer [src] 
2*sin(1)   2*sin(1/2)
-------- + ----------
   3           3
2sin⁡(12)3+2sin⁡(1)3\frac{2 \sin{\left(\frac{1}{2} \right)}}{3} + \frac{2 \sin{\left(1 \right)}}{3} 
=
= 
2*sin(1)   2*sin(1/2)
-------- + ----------
   3           3
2sin⁡(12)3+2sin⁡(1)3\frac{2 \sin{\left(\frac{1}{2} \right)}}{3} + \frac{2 \sin{\left(1 \right)}}{3} 
2*sin(1)/3 + 2*sin(1/2)/3
Numerical answer [src] 
0.880597682274733
0.880597682274733

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

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