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受欢迎的微积分 >

integral of (sec(θ)tan(θ))/(3+2sec(θ))

解答

\int \frac{sec ( θ ) tan ( θ )}{3 + 2 sec ( θ )} d θ

解答

\frac{1}{2} ln ∣ 3 cos (θ) + 2 ∣ - \frac{1}{2} ln (3 ∣ cos (θ) ∣) + C

求解步骤

\int \frac{sec ( θ ) tan ( θ )}{3 + 2 sec ( θ )} d θ

使用三角恒等式改写

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

化简

\frac{sin ( θ )}{cos ( θ ) ( 3 cos ( θ ) + 2 )} : \frac{tan ( θ )}{3 cos ( θ ) + 2}

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

使用换元积分法

= \int - \frac{1}{u ( 3 u + 2 )} d u

提出常数:

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

= - \int \frac{1}{u ( 3 u + 2 )} d u

使用换元积分法

= - \int \frac{1}{v ( v - 2 )} d v

对

v (v - 2)

配方:

(v - 1)^{2} - 1

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

使用换元积分法

= - \int \frac{1}{w ^{2} - 1} d w

因式分解

w^{2} - 1 : - (- w^{2} + 1)

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

提出常数:

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

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

使用积分常见定则:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= - (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

代回

= - (- (\frac{ln ∣ 3 cos ( θ ) + 2 - 1 + 1 ∣}{2} - \frac{ln ∣ 3 cos ( θ ) + 2 - 1 - 1 ∣}{2}))

化简

- (- (\frac{ln ∣ 3 cos ( θ ) + 2 - 1 + 1 ∣}{2} - \frac{ln ∣ 3 cos ( θ ) + 2 - 1 - 1 ∣}{2})) : \frac{1}{2} ln ∣ 3 cos (θ) + 2 ∣ - \frac{1}{2} ln (3 ∣ cos (θ) ∣)

= \frac{1}{2} ln ∣ 3 cos (θ) + 2 ∣ - \frac{1}{2} ln (3 ∣ cos (θ) ∣)

解答补常数

= \frac{1}{2} ln ∣ 3 cos (θ) + 2 ∣ - \frac{1}{2} ln (3 ∣ cos (θ) ∣) + C

作图

流行的例子

integral of (3-0.01x+0.00006x^2)

\int (3 - 0.01 x + 0.00006 x^{2}) d x

integral of 1/((x+2)sqrt(x^2+4x-5))

\int \frac{1}{( x + 2 ) x ^{2} + 4 x - 5} d x

integral of 2cos(x)((2csc(x))/(cot(x)))

\int 2 cos (x) (\frac{2 csc ( x )}{cot ( x )}) d x

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\int \frac{1}{x ( 4 + ( 3 ln ( x ) ) ^{2} )} d x

integral of (e^x)/(1+2e^x+e^{2x)}

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