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Calculus · Definite-Integrals

Integrate -4·sin(t)^2·cos(t) + cos(t) from 0 to 2·pi

\int_{0}^{2 \pi} - 4 \sin^{2}{\left(t \right)} \cos{\left(t \right)} + \cos{\left(t \right)}\, dt

Step by step

  1. \int_{0}^{2 \pi} - 4 \sin^{2}{\left(t \right)} \cos{\left(t \right)} + \cos{\left(t \right)}\, dt

    First find an antiderivative F, then evaluate F(b) − F(a).

  2. \int - 4 \sin^{2}{\left(t \right)} \cos{\left(t \right)} + \cos{\left(t \right)}\, dt = \int - 4 \sin^{2}{\left(t \right)} \cos{\left(t \right)}\, dt + \int \cos{\left(t \right)}\, dt

    The integral of a sum is the sum of the integrals.

  3. \int - 4 \sin^{2}{\left(t \right)} \cos{\left(t \right)}\, dt = -4 \int \sin^{2}{\left(t \right)} \cos{\left(t \right)}\, dt

    Pull the constant -4 out of the integral.

  4. u = \sin{\left(t \right)},\quad du = \cos{\left(t \right)}\, dt

    Substitute u = \sin{\left(t \right)}.

  5. \int \sin^{2}{\left(t \right)} \cos{\left(t \right)}\, dt = \int u^{2}\, d_u

    Rewrite the integral in terms of u.

  6. \int u^{2}\, d_u = \frac{u^{3}}{3}

    Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).

  7. = \frac{\sin^{3}{\left(t \right)}}{3}

    Substitute back u = \sin{\left(t \right)}.

  8. \int \cos{\left(t \right)}\, dt = \sin{\left(t \right)}

    Standard trigonometric antiderivative.

  9. F(2 \pi) - F(0) = \left(0\right) - \left(0\right)

    Fundamental theorem of calculus: plug in the limits.

  10. = 0

    Simplify.

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