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受欢迎的微积分 >

integral of 1/(sin^3(x)cos(x))

解答

\int \frac{1}{sin ^{3} ( x ) cos ( x )} d x

解答

- \frac{1}{2} cot^{2} (x) - ln ∣ cot (x) ∣ + C

求解步骤

\int \frac{1}{sin ^{3} ( x ) cos ( x )} d x

使用三角换元法

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

提出常数:

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

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

化简

(u^{2} + 1)^{\frac{1}{2}} u^{2} + 1 : u^{2} + 1

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

乘开

\frac{u ^{2} + 1}{u} : u + \frac{1}{u}

= - \int u + \frac{1}{u} d u

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

= - (\int u d u + \int \frac{1}{u} d u)

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

\int \frac{1}{u} d u = ln ∣ u ∣

= - (\frac{u ^{2}}{2} + ln ∣ u ∣)

u = cot (x)

代回

= - (\frac{cot ^{2} ( x )}{2} + ln ∣ cot (x) ∣)

化简

- (\frac{cot ^{2} ( x )}{2} + ln ∣ cot (x) ∣) : - \frac{1}{2} cot^{2} (x) - ln ∣ cot (x) ∣

= - \frac{1}{2} cot^{2} (x) - ln ∣ cot (x) ∣

解答补常数

= - \frac{1}{2} cot^{2} (x) - ln ∣ cot (x) ∣ + C

作图

流行的例子

integral of (3x^3)/(1-x^4)

\int \frac{3 x ^{3}}{1 - x ^{4}} d x

integral of (5-2x)/(sqrt(5+4x-x^2))

\int \frac{5 - 2 x}{5 + 4 x - x ^{2}} d x

integral of ((1-x)^2)/(sqrt(x))

\int \frac{( 1 - x ) ^{2}}{x} d x

integral of (cos(2x))/(sin^2(x)cos^2(x))

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

integral of 1/(sqrt(1+x-x^2))

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