integral of (1/t i+j-t^{3/2}k)

解答

\int (\frac{1}{t} i + j - t^{\frac{3}{2}} k) d t

\frac{1}{5} (- 2 k t^{\frac{5}{2}} + 5 j t) + i ln ∣ t ∣ + C

求解步骤

\int \frac{1}{t} i + j - t^{\frac{3}{2}} k d t

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}{t} i d t + \int j d t - \int t^{\frac{3}{2}} k d t

\int \frac{1}{t} i d t = i ln ∣ t ∣

\int j d t = j t

\int t^{\frac{3}{2}} k d t = k \frac{2}{5} t^{\frac{5}{2}}

= i ln ∣ t ∣ + j t - k \frac{2}{5} t^{\frac{5}{2}}

化简

i ln ∣ t ∣ + j t - k \frac{2}{5} t^{\frac{5}{2}} : \frac{1}{5} (- 2 k t^{\frac{5}{2}} + 5 j t) + i ln ∣ t ∣

= \frac{1}{5} (- 2 k t^{\frac{5}{2}} + 5 j t) + i ln ∣ t ∣

解答补常数

= \frac{1}{5} (- 2 k t^{\frac{5}{2}} + 5 j t) + i ln ∣ t ∣ + C

\int 30 sec^{4} (x) d x

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

\int z^{- 4} d z

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

\int (3 x - h)^{3} d x