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

The integral of (1/t i+j-t^{3/2}k) is 1/5 (-2kt^{5/2}+5jt)+iln|t|+C

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

\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

Solution steps

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

Apply the Sum Rule:

\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}}

Simplify

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 ∣

Add a constant to the solution

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

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What is the integral of (1/t i+j-t^{3/2}k) ?
The integral of (1/t i+j-t^{3/2}k) is 1/5 (-2kt^{5/2}+5jt)+iln|t|+C