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人気のある微分積分 >

integral of (t^5)/(sqrt(t^2+10))

解

\int \frac{t ^{5}}{t ^{2} + 10} d t

解

10010 \frac{1}{10} (10 + t^{2}) - \frac{20}{3} (t^{2} + 10)^{\frac{3}{2}} + \frac{1}{5} (10 + t^{2})^{\frac{5}{2}} + C

解答ステップ

\int \frac{t ^{5}}{t ^{2} + 10} d t

三角関数による置換を適用する

= \int 10010 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

= 10010 \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

= 10010 \cdot \int (- 1 + sec^{2} (u))^{2} tan (u) sec (u) d u

u置換積分法を適用する

= 10010 \cdot \int (- 1 + v^{2})^{2} d v

拡張

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

= 10010 \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

= 10010 (\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}

= 10010 (v - \frac{2 v ^{3}}{3} + \frac{v ^{5}}{5})

代用を戻す

= 10010 sec (arctan (\frac{1}{10} t)) - \frac{2 sec ^{3} ( arctan ( \frac{1}{10} t ) )}{3} + \frac{sec ^{5} ( arctan ( \frac{1}{10} t ) )}{5}

簡素化

10010 sec (arctan (\frac{1}{10} t)) - \frac{2 sec ^{3} ( arctan ( \frac{1}{10} t ) )}{3} + \frac{sec ^{5} ( arctan ( \frac{1}{10} t ) )}{5} : 10010 \frac{1}{10} (10 + t^{2}) - \frac{20}{3} (t^{2} + 10)^{\frac{3}{2}} + \frac{1}{5} (10 + t^{2})^{\frac{5}{2}}

= 10010 \frac{1}{10} (10 + t^{2}) - \frac{20}{3} (t^{2} + 10)^{\frac{3}{2}} + \frac{1}{5} (10 + t^{2})^{\frac{5}{2}}

定数を解答に追加する

= 10010 \frac{1}{10} (10 + t^{2}) - \frac{20}{3} (t^{2} + 10)^{\frac{3}{2}} + \frac{1}{5} (10 + t^{2})^{\frac{5}{2}} + C

グラフ

人気の例

(dy)/(dx)=0.5(100-y)

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