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人気のある微分積分 >

integral of sqrt(1+5x^2)

解

\int 1 + 5 x^{2} d x

解

\frac{1}{2} x 1 + 5 x^{2} + \frac{1}{2 5} ln 5 x + 1 + 5 x^{2} + C

解答ステップ

\int 1 + 5 x^{2} d x

三角関数による置換を適用する

= \int \frac{1}{5} sec^{3} (u) d u

定数を除く:

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

= \frac{1}{5} \cdot \int sec^{3} (u) d u

積分換算を適用する

= \frac{1}{5} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

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

= \frac{1}{5} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

代用を戻す

u = arctan (5 x)

= \frac{1}{5} (\frac{sec ^{2} ( arctan ( 5 x ) ) sin ( arctan ( 5 x ) )}{2} + \frac{1}{2} ln tan (arctan (5 x)) + sec (arctan (5 x)))

簡素化

\frac{1}{5} (\frac{sec ^{2} ( arctan ( 5 x ) ) sin ( arctan ( 5 x ) )}{2} + \frac{1}{2} ln tan (arctan (5 x)) + sec (arctan (5 x))) : \frac{1}{2} x 1 + 5 x^{2} + \frac{1}{2 5} ln 5 x + 1 + 5 x^{2}

= \frac{1}{2} x 1 + 5 x^{2} + \frac{1}{2 5} ln 5 x + 1 + 5 x^{2}

定数を解答に追加する

= \frac{1}{2} x 1 + 5 x^{2} + \frac{1}{2 5} ln 5 x + 1 + 5 x^{2} + C

グラフ

人気の例

integral of (3x+2)/((x-4)(x+2))

\int \frac{3 x + 2}{( x - 4 ) ( x + 2 )} d x

integral of (5x+1)(2x-2)

\int (5 x + 1) (2 x - 2) d x

integral of (2x^2-5x-7)/(x-3)

\int \frac{2 x ^{2} - 5 x - 7}{x - 3} d x

integral of (6x)/(e^{3x)}

\int \frac{6 x}{e ^{3 x}} d x

integral of (3+3cot(x))

\int (3 + 3 cot (x)) d x