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人気のある微分積分 >

integral of (1+x^2)^{1/2}

解

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

解

\frac{1}{2} x (x^{2} + 1)^{\frac{1}{2}} + \frac{1}{2} ln x + 1 + x^{2} + C

解答ステップ

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

三角関数による置換を適用する

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

積分換算を適用する

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

代用を戻す

u = arctan (x)

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

簡素化

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

= \frac{1}{2} x (x^{2} + 1)^{\frac{1}{2}} + \frac{1}{2} ln x + 1 + x^{2}

定数を解答に追加する

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

グラフ

人気の例

integral of 2(sin(2x))^2

\int 2 (sin (2 x))^{2} d x

(\partial)/(\partial x)(xy*e^{xy})

\frac{\partial}{\partial x} (x y \cdot e^{x y})

derivative of 3x^2-e^x

\frac{d}{d x} (3 x^{2} - e^{x})

integral of 4^tsin(4^t)

\int 4^{t} sin (4^{t}) d t

derivative of sqrt(y-x+1)+sqrt(x-y+1)

\frac{d}{d x} (y - x + 1 + x - y + 1)