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人気のある微分積分 >

integral of (x^5)/(sqrt(x^2+16))

解

\int \frac{x ^{5}}{x ^{2} + 16} d x

解

1024 (\frac{1}{4} 16 + x^{2} - \frac{1}{96} (16 + x^{2})^{\frac{3}{2}} + \frac{1}{5120} (16 + x^{2})^{\frac{5}{2}}) + C

解答ステップ

\int \frac{x ^{5}}{x ^{2} + 16} d x

三角関数による置換を適用する

= \int 1024 tan^{5} (u) sec (u) d u

定数を除く:

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

= 1024 \cdot \int tan^{5} (u) sec (u) d u

三角関数の公式を使用して書き換える

= 1024 \cdot \int (- 1 + sec^{2} (u))^{2} tan (u) sec (u) d u

u置換積分法を適用する

= 1024 \cdot \int (- 1 + v^{2})^{2} d v

拡張

(- 1 + v^{2})^{2} : 1 - 2 v^{2} + v^{4}

= 1024 \cdot \int 1 - 2 v^{2} + v^{4} d v

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

= 1024 (\int 1 d v - \int 2 v^{2} d v + \int v^{4} d v)

\int 1 d v = v

\int 2 v^{2} d v = \frac{2 v ^{3}}{3}

\int v^{4} d v = \frac{v ^{5}}{5}

= 1024 (v - \frac{2 v ^{3}}{3} + \frac{v ^{5}}{5})

代用を戻す

= 1024 (sec (arctan (\frac{1}{4} x)) - \frac{2 sec ^{3} ( arctan ( \frac{1}{4} x ) )}{3} + \frac{sec ^{5} ( arctan ( \frac{1}{4} x ) )}{5})

簡素化

1024 (sec (arctan (\frac{1}{4} x)) - \frac{2 sec ^{3} ( arctan ( \frac{1}{4} x ) )}{3} + \frac{sec ^{5} ( arctan ( \frac{1}{4} x ) )}{5}) : 1024 (\frac{1}{4} 16 + x^{2} - \frac{1}{96} (16 + x^{2})^{\frac{3}{2}} + \frac{1}{5120} (16 + x^{2})^{\frac{5}{2}})

= 1024 (\frac{1}{4} 16 + x^{2} - \frac{1}{96} (16 + x^{2})^{\frac{3}{2}} + \frac{1}{5120} (16 + x^{2})^{\frac{5}{2}})

定数を解答に追加する

= 1024 (\frac{1}{4} 16 + x^{2} - \frac{1}{96} (16 + x^{2})^{\frac{3}{2}} + \frac{1}{5120} (16 + x^{2})^{\frac{5}{2}}) + C

グラフ

人気の例

integral of 1/(sqrt(4x^2+12x+5))

\int \frac{1}{4 x ^{2} + 12 x + 5} d x

integral of sec^3(5x)tan(5x)

\int sec^{3} (5 x) tan (5 x) d x

integral of (8w+w^2)/((w-7)(w^2+16))

\int \frac{8 w + w ^{2}}{( w - 7 ) ( w ^{2} + 16 )} d w

integral of sqrt(x)(sqrt(1+\sqrt{x)})

\int x (1 + x) d x

integral of ((x^2+x))/(x^3-x^2+x-1)

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