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人気のある微分積分 >

integral of (2sqrt(x^2-1))/x

解

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

解

2 (- arctan (x^{2} - 1) + x^{2} - 1) + C

解答ステップ

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

定数を除く:

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

= 2 \cdot \int \frac{x ^{2} - 1}{x} d x

u置換積分法を適用する

= 2 \cdot \int \frac{u ^{2}}{u ^{2} + 1} d u

\frac{u ^{2}}{u ^{2} + 1} = - \frac{1}{u ^{2} + 1} + 1

= 2 \cdot \int - \frac{1}{u ^{2} + 1} + 1 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

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

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

\int 1 d u = u

= 2 (- arctan (u) + u)

代用を戻す

u = x^{2} - 1

= 2 (- arctan (x^{2} - 1) + x^{2} - 1)

定数を解答に追加する

= 2 (- arctan (x^{2} - 1) + x^{2} - 1) + C

グラフ

人気の例

integral of (1+3z)(2sqrt(z)-5)

\int (1 + 3 z) (2 z - 5) d z

integral of (11x^2)/((81+x^2)^2)

\int \frac{11 x ^{2}}{( 81 + x ^{2} ) ^{2}} d x

integral of (2x^2+1)/((x+3)(x+2))

\int \frac{2 x ^{2} + 1}{( x + 3 ) ( x + 2 )} d x

integral of (sec^2(y))/(1+tan(y))

\int \frac{sec ^{2} ( y )}{1 + tan ( y )} d y

integral of (cos^7(x))/(sqrt(sin(x)))

\int \frac{cos ^{7} ( x )}{sin ( x )} d x