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चरण द्वारा कदम

1. \int_{- \pi}^{\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(- \pi) = \left(0\right) - \left(0\right)

   Fundamental theorem of calculus: plug in the limits.

8. = 0

   Simplify.