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受欢迎的微积分 >

integral of (cos(θ))/(sqrt(2+sin^2(θ)))

解答

\int \frac{cos ( θ )}{2 + sin ^{2} ( θ )} d θ

解答

ln \frac{1}{2} sin (θ) + \frac{1}{2} (2 + sin^{2} (θ)) + C

求解步骤

\int \frac{cos ( θ )}{2 + sin ^{2} ( θ )} d θ

使用三角恒等式改写

= \int \frac{cos ( θ )}{- cos ^{2} ( θ ) + 3} d θ

使用换元积分法

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

使用三角换元法

= \int sec (v) d v

使用积分常见定则:

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= ln ∣ tan (v) + sec (v) ∣

代回

= ln tan (arctan (\frac{1}{2} sin (θ))) + sec (arctan (\frac{1}{2} sin (θ)))

化简

ln tan (arctan (\frac{1}{2} sin (θ))) + sec (arctan (\frac{1}{2} sin (θ))) : ln \frac{1}{2} sin (θ) + \frac{1}{2} (2 + sin^{2} (θ))

= ln \frac{1}{2} sin (θ) + \frac{1}{2} (2 + sin^{2} (θ))

解答补常数

= ln \frac{1}{2} sin (θ) + \frac{1}{2} (2 + sin^{2} (θ)) + C

作图

流行的例子

limit as x approaches 0 of (4x^2)/x

x \to 0 lim (\frac{4 x ^{2}}{x})

(\partial)/(\partial x)((41-x^2-y^2)^{1/2})

\frac{\partial}{\partial x} ((41 - x^{2} - y^{2})^{\frac{1}{2}})

integral from 0 to 1 of xe^{-4x}

\int_{0}^{1} x e^{- 4 x} d x

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\frac{d}{d x} (3 e^{2 x} + 2 e^{2 x} x)

derivative (x^6)/6+1/(16x^4)

d er i v a t i v e \frac{x ^{6}}{6} + \frac{1}{16 x ^{4}}