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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

u置換積分法を適用する

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

定数を除く:

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

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

平方完成する

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

u置換積分法を適用する

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

定数を除く:

\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

置換積分法を適用する

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

定数を除く:

\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

共通積分を使用する:

\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})

代用を戻す

= 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})

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

derivative of e^{-x^2-y^2}cos(xy)

\frac{d}{d x} (e^{- x^{2} - y^{2}} cos (x y))

sum from n=1 to infinity of n^2e^{-n^3}

n = 1 \sum \infty n^{2} e^{- n^{3}}

inverselaplace 6/(s^2+9)

in v erse l a pl a ce \frac{6}{s ^{2} + 9}

limit as x approaches 3-of 2/((x-3))

x \to 3 - lim (\frac{2}{( x - 3 )})

(\partial)/(\partial u(v))((e^v)/(u(v)+v^9))

\frac{\partial}{\partial u ( v )} (\frac{e ^{v}}{u ( v ) + v ^{9}})