积分 ((1+x^2))/((x^3))

解答

积分

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

- \frac{1}{2 x ^{2}} + ln ∣ x ∣ + C

求解步骤

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

乘开

\frac{1 + x ^{2}}{x ^{3}} : \frac{1}{x ^{3}} + \frac{1}{x}

= \int \frac{1}{x ^{3}} + \frac{1}{x} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x ^{3}} d x + \int \frac{1}{x} d x

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

\int \frac{1}{x} d x = ln ∣ x ∣

= - \frac{1}{2 x ^{2}} + ln ∣ x ∣

解答补常数

= - \frac{1}{2 x ^{2}} + ln ∣ x ∣ + C

in t e g r a l \frac{( x - 1 )}{( x + 1 )}

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

in t e g r a l \frac{t}{( t + 2 )}

in t e g r a l \frac{tan ( x )}{( 1 + 3 tan ^{2} ( x ) )}

in t e g r a l lo g_{10} (a^{2} + x^{2})