integral from-2pi to-pi of cos^2(x)

解答

\int_{- 2 π}^{- π} cos^{2} (x) d x

\frac{π}{2}

十进制

1.57079 \dots

求解步骤

\int_{- 2 π}^{- π} cos^{2} (x) d x

使用三角恒等式改写

= \int_{- 2 π}^{- 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_{- 2 π}^{- 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_{- 2 π}^{- π} 1 d x + \int_{- 2 π}^{- π} cos (2 x) d x)

\int_{- 2 π}^{- π} 1 d x = π

\int_{- 2 π}^{- 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}

\frac{d}{d x} ((- sin (x))^{2})

x \to 0 lim (x^{2} + 5)

\int \frac{1}{2 - 5 x ^{2}} d x

\frac{d}{d x} (\frac{( sin ( x ) )}{1 + cos ( x )})

x \to 0 + lim (x ln (3 x))