ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of x^2(9-x^2)^{3/2}

解

\int x^{2} (9 - x^{2})^{\frac{3}{2}} d x

解

729 (\frac{1}{81} x (9 - x^{2})^{\frac{3}{2}} + \frac{3}{8} arcsin (\frac{1}{3} x) - \frac{3}{32} sin (4 arcsin (\frac{1}{3} x)) - \frac{1}{4374} x (9 - x^{2})^{\frac{5}{2}} - \frac{5}{1944} x (9 - x^{2})^{\frac{3}{2}} - \frac{5}{32} (2 arcsin (\frac{1}{3} x) + sin (2 arcsin (\frac{1}{3} x)))) + C

解答ステップ

\int x^{2} (9 - x^{2})^{\frac{3}{2}} d x

三角関数による置換を適用する

= \int 729 cos^{4} (u) sin^{2} (u) d u

定数を除く:

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

= 729 \cdot \int cos^{4} (u) sin^{2} (u) d u

三角関数の公式を使用して書き換える

= 729 \cdot \int cos^{4} (u) (1 - cos^{2} (u)) d u

拡張

cos^{4} (u) (1 - cos^{2} (u)) : cos^{4} (u) - cos^{6} (u)

= 729 \cdot \int cos^{4} (u) - cos^{6} (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

= 729 (\int cos^{4} (u) d u - \int cos^{6} (u) d u)

\int cos^{4} (u) d u = cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u))

\int cos^{6} (u) d u = \frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))

= 729 (cos^{3} (u) sin (u) + \frac{3}{8} (u - \frac{1}{4} sin (4 u)) - (\frac{sin ( u ) cos ^{5} ( u )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (u) sin (u) + \frac{3}{8} (u + \frac{1}{2} sin (2 u)))))

代用を戻す

u = arcsin (\frac{1}{3} x)

= 729 (cos^{3} (arcsin (\frac{1}{3} x)) sin (arcsin (\frac{1}{3} x)) + \frac{3}{8} (arcsin (\frac{1}{3} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{3} x))) - (\frac{sin ( arcsin ( \frac{1}{3} x ) ) cos ^{5} ( arcsin ( \frac{1}{3} x ) )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (arcsin (\frac{1}{3} x)) sin (arcsin (\frac{1}{3} x)) + \frac{3}{8} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))))))

簡素化

729 (cos^{3} (arcsin (\frac{1}{3} x)) sin (arcsin (\frac{1}{3} x)) + \frac{3}{8} (arcsin (\frac{1}{3} x) - \frac{1}{4} sin (4 arcsin (\frac{1}{3} x))) - (\frac{sin ( arcsin ( \frac{1}{3} x ) ) cos ^{5} ( arcsin ( \frac{1}{3} x ) )}{6} + \frac{5}{6} (\frac{1}{4} cos^{3} (arcsin (\frac{1}{3} x)) sin (arcsin (\frac{1}{3} x)) + \frac{3}{8} (arcsin (\frac{1}{3} x) + \frac{1}{2} sin (2 arcsin (\frac{1}{3} x)))))) : 729 (\frac{1}{81} x (9 - x^{2})^{\frac{3}{2}} + \frac{3}{8} arcsin (\frac{1}{3} x) - \frac{3}{32} sin (4 arcsin (\frac{1}{3} x)) - \frac{1}{4374} x (9 - x^{2})^{\frac{5}{2}} - \frac{5}{1944} x (9 - x^{2})^{\frac{3}{2}} - \frac{5}{32} (2 arcsin (\frac{1}{3} x) + sin (2 arcsin (\frac{1}{3} x))))

= 729 (\frac{1}{81} x (9 - x^{2})^{\frac{3}{2}} + \frac{3}{8} arcsin (\frac{1}{3} x) - \frac{3}{32} sin (4 arcsin (\frac{1}{3} x)) - \frac{1}{4374} x (9 - x^{2})^{\frac{5}{2}} - \frac{5}{1944} x (9 - x^{2})^{\frac{3}{2}} - \frac{5}{32} (2 arcsin (\frac{1}{3} x) + sin (2 arcsin (\frac{1}{3} x))))

定数を解答に追加する

= 729 (\frac{1}{81} x (9 - x^{2})^{\frac{3}{2}} + \frac{3}{8} arcsin (\frac{1}{3} x) - \frac{3}{32} sin (4 arcsin (\frac{1}{3} x)) - \frac{1}{4374} x (9 - x^{2})^{\frac{5}{2}} - \frac{5}{1944} x (9 - x^{2})^{\frac{3}{2}} - \frac{5}{32} (2 arcsin (\frac{1}{3} x) + sin (2 arcsin (\frac{1}{3} x)))) + C

グラフ

人気の例

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

\int \frac{5 x}{( x ^{2} + 1 ) ( x - 2 )} d x

integral of (e^{9/x})/(x^2)

\int \frac{e ^{\frac{9}{x}}}{x ^{2}} d x

integral of 64xsec^2(4x)

\int 64 x sec^{2} (4 x) d x

integral of z/((a^2+z^2)^{3/2)}

\int \frac{z}{( a ^{2} + z ^{2} ) ^{\frac{3}{2}}} d z

integral of ((5x^2+10x-30))/(x^2+x-6)

\int \frac{( 5 x ^{2} + 10 x - 30 )}{x ^{2} + x - 6} d x