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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

三角関数による置換を適用する

= \int \frac{cos ( u )}{1 + cos ( u )} d u

1 + cos (u)

の共役で乗じる

\frac{sin ^{2} ( u )}{1 - cos ( u )}

= \int \frac{cos ( u )}{\frac{s i n ^{2} ( u )}{1 - c o s ( u )}} d u

拡張

\frac{cos ( u )}{\frac{s i n ^{2} ( u )}{1 - c o s ( u )}} : \frac{cos ( u )}{sin ^{2} ( u )} - \frac{cos ^{2} ( u )}{sin ^{2} ( u )}

= \int \frac{cos ( u )}{sin ^{2} ( u )} - \frac{cos ^{2} ( u )}{sin ^{2} ( u )} d u

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

= \int \frac{cos ( u )}{sin ^{2} ( u )} d u - \int \frac{cos ^{2} ( u )}{sin ^{2} ( u )} d u

\int \frac{cos ( u )}{sin ^{2} ( u )} d u = - csc (u)

\int \frac{cos ^{2} ( u )}{sin ^{2} ( u )} d u = - cot (u) - u

= - csc (u) - (- cot (u) - u)

代用を戻す

u = arcsin (x)

= - csc (arcsin (x)) - (- cot (arcsin (x)) - arcsin (x))

簡素化

- csc (arcsin (x)) - (- cot (arcsin (x)) - arcsin (x)) : - \frac{1}{x} + \frac{1 - x ^{2}}{x} + arcsin (x)

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

定数を解答に追加する

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

グラフ

人気の例

integral of ysqrt(9-y^2)

\int y 9 - y^{2} d y

integral of 3/(9x^2+4)

\int \frac{3}{9 x ^{2} + 4} d x

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

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

integral of 1/(120-x)

\int \frac{1}{120 - x} d x

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

\int \frac{e ^{2 x} + 1}{e ^{2 x} - 2 e ^{x} + 1} d x