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受欢迎的微积分 >

integral of 1/(3cos(x)-4sin(x))

解答

\int \frac{1}{3 cos ( x ) - 4 sin ( x )} d x

解答

\frac{1}{5} (ln \frac{9}{5} + \frac{3}{5} tan (\frac{x}{2}) - ln - \frac{1}{5} + \frac{3}{5} tan (\frac{x}{2})) + C

求解步骤

\int \frac{1}{3 cos ( x ) - 4 sin ( x )} d x

因式分解

3 cos (x) - 4 sin (x) : - (- 3 cos (x) + 4 sin (x))

= \int \frac{1}{- ( - 3 cos ( x ) + 4 sin ( x ) )} d x

提出常数:

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

= - \int \frac{1}{- 3 cos ( x ) + 4 sin ( x )} d x

使用换元积分法

= - \int \frac{2}{- 3 ( 1 - u ^{2} ) + 8 u} d u

提出常数:

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

= - 2 \cdot \int \frac{1}{- 3 ( 1 - u ^{2} ) + 8 u} d u

对

- 3 (1 - u^{2}) + 8 u

配方:

3 (u + \frac{4}{3})^{2} - \frac{25}{3}

= - 2 \cdot \int \frac{1}{3 ( u + \frac{4}{3} ) ^{2} - \frac{25}{3}} d u

使用换元积分法

= - 2 \cdot \int \frac{3}{9 v ^{2} - 25} d v

提出常数:

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

= - 2 \cdot 3 \cdot \int \frac{1}{9 v ^{2} - 25} d v

使用换元积分法

= - 2 \cdot 3 \cdot \int \frac{1}{15 ( w ^{2} - 1 )} d w

提出常数:

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

= - 2 \cdot 3 \cdot \frac{1}{15} \cdot \int \frac{1}{w ^{2} - 1} d w

因式分解

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

= - 2 \cdot 3 \cdot \frac{1}{15} \cdot \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

= - 2 \cdot 3 \cdot \frac{1}{15} (- \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}

= - 2 \cdot 3 \cdot \frac{1}{15} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

代回

= - 2 \cdot 3 \cdot \frac{1}{15} (- (\frac{ln \frac{3}{5} ( tan ( \frac{x}{2} ) + \frac{4}{3} ) + 1}{2} - \frac{ln \frac{3}{5} ( tan ( \frac{x}{2} ) + \frac{4}{3} ) - 1}{2}))

化简

- 2 \cdot 3 \cdot \frac{1}{15} (- (\frac{ln \frac{3}{5} ( tan ( \frac{x}{2} ) + \frac{4}{3} ) + 1}{2} - \frac{ln \frac{3}{5} ( tan ( \frac{x}{2} ) + \frac{4}{3} ) - 1}{2})) : \frac{1}{5} (ln \frac{9}{5} + \frac{3}{5} tan (\frac{x}{2}) - ln - \frac{1}{5} + \frac{3}{5} tan (\frac{x}{2}))

= \frac{1}{5} (ln \frac{9}{5} + \frac{3}{5} tan (\frac{x}{2}) - ln - \frac{1}{5} + \frac{3}{5} tan (\frac{x}{2}))

解答补常数

= \frac{1}{5} (ln \frac{9}{5} + \frac{3}{5} tan (\frac{x}{2}) - ln - \frac{1}{5} + \frac{3}{5} tan (\frac{x}{2})) + C

作图

流行的例子

limit as x approaches-8-of (x+3)/(x+8)

x \to - 8 - lim (\frac{x + 3}{x + 8})

integral of 9/(x(x^4+1))

\int \frac{9}{x ( x ^{4} + 1 )} d x

integral of 2x^{25}

\int 2 x^{25} d x

limit as n approaches infinity of \sqrt[n]{n^3}

n \to \infty lim (n n^{3})

derivative f(x)=x^4-2x^3-12x^2+4x-3

d er i v a t i v e f (x) = x^{4} - 2 x^{3} - 12 x^{2} + 4 x - 3