Integrating a product, one factor a derivative \int \frac{dx(t)}{dt}x(t)^{2}dt

Betsy Rhone

Betsy Rhone

Answered question

2022-01-21

Integrating a product, one factor a derivative
dx(t)dtx(t)2dt

Answer & Explanation

Elois Puryear

Elois Puryear

Beginner2022-01-21Added 30 answers

If integration by parts gave you 0, its
Dabanka4v

Dabanka4v

Beginner2022-01-22Added 36 answers

I’m going to assume that x(t)dt is a typo for d(x(t))dt.
Look at a specific example, say with x(t)=sint. Then
d(x(t))dtx(t)2dt=costsin2tdt,
a problem that you would most likely solve by making the substitution u=sint,du=costdt, and integrating
u2du.
But you don’t have to know what x(t) is to make this substitution. If you let u=x(t), then
du=d(x(t))dtdt,
and d(x(t))dtx(t)2dt=u2du=u33+C=x(t)33+C
RizerMix

RizerMix

Expert2022-01-27Added 656 answers

Also, since dx(t)dtx(t)2dt=df(t)dtdt, where f(t)=13x(t)3, we can use the Second Fundamental Theorem of Calculus to conclude that dx(t)dtx(t)2dt=f(t)+C=13x(t)3+C. No substitution needed.

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