zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of ((x^2-16)^{3/2})/(x^3)

解答

\int \frac{( x ^{2} - 16 ) ^{\frac{3}{2}}}{x ^{3}} d x

解答

\frac{16 ( x ^{2} - 16 ) ^{\frac{3}{2}}}{x ^{4}} - 6 arcsec (\frac{1}{4} x) + 3 sin (2 arcsec (\frac{1}{4} x)) + \frac{( x ^{2} - 16 ) ^{\frac{5}{2}}}{x ^{4}} + C

求解步骤

\int \frac{( x ^{2} - 16 ) ^{\frac{3}{2}}}{x ^{3}} d x

使用三角换元法

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

提出常数:

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

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

使用三角恒等式改写

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

使用分布积分法

= 4 (- \int 4 sin^{4} (u) d u + sin^{4} (u) tan (u))

\int 4 sin^{4} (u) d u = - sin^{3} (u) cos (u) + \frac{3}{2} u - \frac{3}{4} sin (2 u)

= 4 (- (- sin^{3} (u) cos (u) + \frac{3}{2} u - \frac{3}{4} sin (2 u)) + sin^{4} (u) tan (u))

u = arcsec (\frac{1}{4} x)

代回

= 4 (- (- sin^{3} (arcsec (\frac{1}{4} x)) cos (arcsec (\frac{1}{4} x)) + \frac{3}{2} arcsec (\frac{1}{4} x) - \frac{3}{4} sin (2 arcsec (\frac{1}{4} x))) + sin^{4} (arcsec (\frac{1}{4} x)) tan (arcsec (\frac{1}{4} x)))

化简

4 (- (- sin^{3} (arcsec (\frac{1}{4} x)) cos (arcsec (\frac{1}{4} x)) + \frac{3}{2} arcsec (\frac{1}{4} x) - \frac{3}{4} sin (2 arcsec (\frac{1}{4} x))) + sin^{4} (arcsec (\frac{1}{4} x)) tan (arcsec (\frac{1}{4} x))) : \frac{16 ( x ^{2} - 16 ) ^{\frac{3}{2}}}{x ^{4}} - 6 arcsec (\frac{1}{4} x) + 3 sin (2 arcsec (\frac{1}{4} x)) + \frac{( x ^{2} - 16 ) ^{\frac{5}{2}}}{x ^{4}}

= \frac{16 ( x ^{2} - 16 ) ^{\frac{3}{2}}}{x ^{4}} - 6 arcsec (\frac{1}{4} x) + 3 sin (2 arcsec (\frac{1}{4} x)) + \frac{( x ^{2} - 16 ) ^{\frac{5}{2}}}{x ^{4}}

解答补常数

= \frac{16 ( x ^{2} - 16 ) ^{\frac{3}{2}}}{x ^{4}} - 6 arcsec (\frac{1}{4} x) + 3 sin (2 arcsec (\frac{1}{4} x)) + \frac{( x ^{2} - 16 ) ^{\frac{5}{2}}}{x ^{4}} + C

作图

流行的例子

integral of 6x^3

\int 6 x^{3} d x

integral of x7^x

\int x 7^{x} d x

integral of (sin(2x))/(4+cos^2(x))

\int \frac{sin ( 2 x )}{4 + cos ^{2} ( x )} d x

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

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

integral of sin(9x)cos(6x)

\int sin (9 x) cos (6 x) d x