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人気のある微分積分 >

integral of (4sqrt(ln^2(x)+49))/x

解

\int \frac{4 ln ^{2} ( x ) + 49}{x} d x

解

196 (\frac{1}{98} ln (x) 49 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{7} ln (x) + 49 + ln^{2} (x))) + C

解答ステップ

\int \frac{4 ln ^{2} ( x ) + 49}{x} d x

定数を除く:

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

= 4 \cdot \int \frac{ln ^{2} ( x ) + 49}{x} d x

u置換積分法を適用する

= 4 \cdot \int u^{2} + 49 d u

三角関数による置換を適用する

= 4 \cdot \int 49 sec^{3} (v) d v

定数を除く:

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

= 4 \cdot 49 \cdot \int sec^{3} (v) d v

積分換算を適用する

= 4 \cdot 49 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} \cdot \int sec (v) d v)

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= 4 \cdot 49 (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代用を戻す

= 4 \cdot 49 (\frac{sec ^{2} ( arctan ( \frac{1}{7} ln ( x ) ) ) sin ( arctan ( \frac{1}{7} ln ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{7} ln (x))) + sec (arctan (\frac{1}{7} ln (x))))

簡素化

4 \cdot 49 (\frac{sec ^{2} ( arctan ( \frac{1}{7} ln ( x ) ) ) sin ( arctan ( \frac{1}{7} ln ( x ) ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{7} ln (x))) + sec (arctan (\frac{1}{7} ln (x)))) : 196 (\frac{1}{98} ln (x) 49 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{7} ln (x) + 49 + ln^{2} (x)))

= 196 (\frac{1}{98} ln (x) 49 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{7} ln (x) + 49 + ln^{2} (x)))

定数を解答に追加する

= 196 (\frac{1}{98} ln (x) 49 + ln^{2} (x) + \frac{1}{2} ln (\frac{1}{7} ln (x) + 49 + ln^{2} (x))) + C

グラフ

人気の例

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