How to integrate of the derivative of a variable?

How to calculate integrals of the type

$$\int \frac{d\phi}{dt}d\phi,$$

$$\int \phi \, d\left(\frac{d\phi}{dt}\right)$$

and

$$\int \sin\left(\frac{d\theta}{dt}\right)\,d\theta$$ ?

Thanks

1 Answer

In each case there needs to be a known relationship between the variables in order to perform the integration.

In the first case, if $\phi=\phi(t)$ then $$I_1=\int\left(\frac{d\phi}{dt}\right)^2\,dt$$

likewise in the second case, $$I_2=\int\phi\frac{d^2\phi}{dt^2}\,dt$$

In the third case, provided $\theta=\theta(t),$ $$I_3=\int\sin\left(\frac{d\theta}{dt}\right)\frac{d\theta}{dt}\,dt$$

• Just wondering if you could expand a bit more on this, and if you could explain what kind of rule/substitution you used in your answer. Aug 15, 2016 at 21:42
• Parametric integration in the first case, i.e. $dx=\frac{dx}{dt}dt$, Integration by substitution in the second and third cases. Aug 15, 2016 at 21:46
• Thanks, that makes more sense now. Aug 15, 2016 at 21:47