integral from 0 to pi of sin(x)sin(x)

解答

\int_{0}^{π} sin (x) sin (x) d x

\frac{π}{2}

十进制

1.57079 \dots

求解步骤

\int_{0}^{π} sin (x) sin (x) d x

化简

sin (x) sin (x) : sin^{2} (x)

= \int_{0}^{π} sin^{2} (x) d x

使用三角恒等式改写

= \int_{0}^{π} \frac{1 - cos ( 2 x )}{2} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{0}^{π} 1 - cos (2 x) d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{0}^{π} 1 d x - \int_{0}^{π} cos (2 x) d x)

\int_{0}^{π} 1 d x = π

\int_{0}^{π} cos (2 x) d x = 0

= \frac{1}{2} (π - 0)

化简

\frac{1}{2} (π - 0) : \frac{π}{2}

= \frac{π}{2}

\int_{- 1}^{2} x^{2} - x d x

\int_{0}^{2} 4 a x e^{a x^{2}} d x

\int_{0}^{1} k (1 - y) y^{n} d y

\int_{0}^{3} ∣ 2 x - 2 ∣ d x

\int_{2}^{4} t^{2} - 2 t - 3 d t