Anonymous
Anonymous asked in Science & MathematicsMathematics · 3 weeks ago

Calc integration help?

integral of xsinx^2(cosx^2)^8

2 Answers

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  • 3 weeks ago

    Look inside the outermost "messy part" of the integrand for a possible substitution.

    Inside of (cos x²)^8 is cos x², so try that.  Let u = cos x², and the chain rule gives:

        du/dx = -2x sin x²

    That will work, making (cos x²)^8 = u^8 and x sin x² dx = - (1/2) du.  So:

       ∫ x sin x² (cos x²)^8 dx = -(1/2) ∫ u^8 du = (-1/18) u^9 + C = (-1/18) (cos x²)^9 + C

  • When integrating, always include the differential.  It's important

    x * sin(x^2) * cos(x^2)^8 * dx

    Use a substitution

    a = x^2

    da = 2x * dx

    x * sin(x^2) * cos(x^2)^8 * dx =>

    (1/2) * sin(x^2) * cos(x^2)^8 * 2x * dx =>

    (1/2) * sin(a) * cos(a)^8 * da =>

    (1/2) * cos(a)^8 * sin(a) * da

    One more substitution

    u = cos(a)

    du = -sin(a) * da

    (1/2) * cos(a)^8 * sin(a) * da =>

    (-1/2) * cos(a)^8 * (-sin(a) * da) =>

    (-1/2) * u^8 * du

    Now we can integrate.

    (-1/2) * (1/9) * u^9 + C =>

    (-1/18) * cos(a)^9 + C =>

    (-1/18) * cos(x^2)^9 + C

    There you go.

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