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Calculus · Definite-Integrals

Integrate sin(x)·sin(2x) from 0 to pi

\int_{0}^{\pi} \sin{\left(x \right)} \sin{\left(2 x \right)}\, dx

Step by step

  1. \int_{0}^{\pi} \sin{\left(x \right)} \sin{\left(2 x \right)}\, dx

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

  2. \int 2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx

    Pull the constant 2 out of the integral.

  3. u = \sin{\left(x \right)},\quad du = \cos{\left(x \right)}\, dx

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

  4. \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx = \int u^{2}\, d_u

    Rewrite the integral in terms of u.

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

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

  6. = \frac{\sin^{3}{\left(x \right)}}{3}

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

  7. F(\pi) - F(0) = \left(0\right) - \left(0\right)

    Fundamental theorem of calculus: plug in the limits.

  8. = 0

    Simplify.

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