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人気のある微分積分 >

integral of (11x^2)/((144+x^2)^2)

解

\int \frac{11 x ^{2}}{( 144 + x ^{2} ) ^{2}} d x

解

\frac{11 ( x ^{2} arctan ( \frac{x}{12} ) - 12 x + 144 arctan ( \frac{x}{12} ) )}{24 ( 144 + x ^{2} )} + C

解答ステップ

\int \frac{11 x ^{2}}{( 144 + x ^{2} ) ^{2}} d x

定数を除く:

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

= 11 \cdot \int \frac{x ^{2}}{( 144 + x ^{2} ) ^{2}} d x

置換積分法を適用する

= 11 \cdot \int \frac{u ^{2}}{12 ( u ^{2} + 1 ) ^{2}} d u

定数を除く:

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

= 11 \cdot \frac{1}{12} \cdot \int \frac{u ^{2}}{( u ^{2} + 1 ) ^{2}} d u

以下の部分分数を得る:

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

= 11 \cdot \frac{1}{12} \cdot \int \frac{1}{u ^{2} + 1} - \frac{1}{( u ^{2} + 1 ) ^{2}} d u

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 11 \cdot \frac{1}{12} (\int \frac{1}{u ^{2} + 1} d u - \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u)

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

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

= 11 \cdot \frac{1}{12} (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

代用を戻す

u = \frac{x}{12}

= 11 \cdot \frac{1}{12} (arctan (\frac{x}{12}) - \frac{1}{2} (arctan (\frac{x}{12}) + \frac{\frac{x}{12}}{1 + ( \frac{x}{12} ) ^{2}}))

簡素化

11 \cdot \frac{1}{12} (arctan (\frac{x}{12}) - \frac{1}{2} (arctan (\frac{x}{12}) + \frac{\frac{x}{12}}{1 + ( \frac{x}{12} ) ^{2}})) : \frac{11 ( x ^{2} arctan ( \frac{x}{12} ) - 12 x + 144 arctan ( \frac{x}{12} ) )}{24 ( 144 + x ^{2} )}

= \frac{11 ( x ^{2} arctan ( \frac{x}{12} ) - 12 x + 144 arctan ( \frac{x}{12} ) )}{24 ( 144 + x ^{2} )}

定数を解答に追加する

= \frac{11 ( x ^{2} arctan ( \frac{x}{12} ) - 12 x + 144 arctan ( \frac{x}{12} ) )}{24 ( 144 + x ^{2} )} + C

グラフ

人気の例

derivative of x^6ln(7x)

\frac{d}{d x} (x^{6} ln (7 x))

derivative y= 1/((x^2-1)(x^2+3x+3))

d er i v a t i v e y = \frac{1}{( x ^{2} - 1 ) ( x ^{2} + 3 x + 3 )}

derivative of x^{9/4}

\frac{d}{d x} (x^{\frac{9}{4}})

derivative f(θ)=sqrt(11θ)

d er i v a t i v e f (θ) = 11 θ

f(s)=sqrt((s^2+1)/(s^2+4))

f (s) = \frac{s ^{2} + 1}{s ^{2} + 4}