integral from 0 to pi/2 of (cos(t))/(sqrt(1+sin^2(t)))

解答

\int_{0}^{\frac{π}{2}} \frac{cos ( t )}{1 + sin ^{2} ( t )} d t

ln (1 + 2)

十进制

0.88137 \dots

求解步骤

\int_{0}^{\frac{π}{2}} \frac{cos ( t )}{1 + sin ^{2} ( t )} d t

使用换元积分法

= \int_{0}^{1} \frac{1}{1 + u ^{2}} d u

使用积分常见定则:

\int \frac{1}{1 + u ^{2}} d u = ln u^{2} + 1 + u

= [ln u^{2} + 1 + u]_{0}^{1}

带入上下限计算后得:

ln (1 + 2)

= ln (1 + 2)

\int \frac{10}{x ^{2} x ^{2} + 16} d x

\int \frac{1}{x + 2} d x

x \to - 3 lim (- x + 6 - 3)

d er i v a t i v e (3 x^{2} + 1)^{2}

\frac{d ^{5}}{d x ^{5}} (\frac{1}{x - 1})