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Calculus · Definite-Integrals

Integrate x·sin(3x) from -pi to pi

\int_{- \pi}^{\pi} x \sin{\left(3 x \right)}\, dx

Step by step

  1. \int_{- \pi}^{\pi} x \sin{\left(3 x \right)}\, dx

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

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

    Integration by parts: ∫u dv = uv − ∫v du.

  3. du = 1\, dx,\quad v = - \frac{\cos{\left(3 x \right)}}{3}

    Differentiate u, integrate dv.

  4. \int x \sin{\left(3 x \right)}\, dx = - \frac{x \cos{\left(3 x \right)}}{3} - \int - \frac{\cos{\left(3 x \right)}}{3}\, dx

    Apply the formula.

  5. \int - \frac{\cos{\left(3 x \right)}}{3}\, dx = - \frac{1}{3} \int \cos{\left(3 x \right)}\, dx

    Pull the constant - \frac{1}{3} out of the integral.

  6. u = 3 x,\quad du = 3\, dx

    Substitute u = 3 x.

  7. \int \cos{\left(3 x \right)}\, dx = \int \frac{\cos{\left(u \right)}}{3}\, d_u

    Rewrite the integral in terms of u.

  8. \int \frac{\cos{\left(u \right)}}{3}\, d_u = \frac{1}{3} \int \cos{\left(u \right)}\, d_u

    Pull the constant \frac{1}{3} out of the integral.

  9. \int \cos{\left(u \right)}\, d_u = \sin{\left(u \right)}

    Standard trigonometric antiderivative.

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

    Substitute back u = 3 x.

  11. = - \frac{x \cos{\left(3 x \right)}}{3} + \frac{\sin{\left(3 x \right)}}{9}

    Combine.

  12. F(\pi) - F(- \pi) = \left(\frac{\pi}{3}\right) - \left(- \frac{\pi}{3}\right)

    Fundamental theorem of calculus: plug in the limits.

  13. = \frac{2 \pi}{3} \approx 2.0944

    Simplify.

Reveal the answer
\frac{2 \pi}{3}