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Solve the integral integral (2x) dx

Mathematics

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Answer

12sin⁡(2x)+C\frac{1}{2}\sin(2x) + C

Step-by-step solution

6 steps

To solve the integral ∫cos⁡(2x)dx\int \cos(2x) dx, we will use a substitution method.

  1. Set up the substitution.

    Let u=2xu = 2x.

  2. Find the differential dudu.

    Differentiate uu with respect to xx:

    dudx=ddx(2x)=2\frac{du}{dx} = \frac{d}{dx}(2x) = 2

    Rearrange to solve for dxdx:

    dx=du2dx = \frac{du}{2}

  3. Substitute uu and dxdx into the integral.

    ∫cos⁡(u)(du2)\int \cos(u) \left(\frac{du}{2}\right)

  4. Pull the constant out of the integral.

    12∫cos⁡(u)du\frac{1}{2} \int \cos(u) du

  5. Integrate cos⁡(u)\cos(u) with respect to uu.

    The integral of cos⁡(u)\cos(u) is sin⁡(u)\sin(u). Remember to add the constant of integration, CC.

    12(sin⁡(u)+C)\frac{1}{2} (\sin(u) + C)

  6. Substitute back u=2xu = 2x.

    12sin⁡(2x)+C\frac{1}{2} \sin(2x) + C
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