How Do You Integrate Trigonometric Vectors with Variable Substitution?

[afcwestwarrior]

3 sin ^2 t cos t (i) + 3 sin t cos^2 t (j) + 2 sin t cost (k)


I have to take the antiderivate for each Vector.

Then I have to evaluate it from pi/2 to o.

I'm confused because I can't use a trigonometric substitution.

Cosine is odd for the I vector but I can't substitute 1-sin^2 t in for it. Because it has to be cos^2 t

 

[rock.freak667]

for the i component, try a substitution since d/dt(sint)=cost, similar with the other two components.

 

[afcwestwarrior]

I get it. If I make it like this, Sin (t) * Sin (t) * Cos (t) dt

t= sin (t)
dt = cos t)

it becomes t*t dt

then it becomes t^2 dt

and the integral of this is t^3 / 3

 

[Mark44]

I get it. If I make it like this, Sin (t) * Sin (t) * Cos (t) dt

t= sin (t)
dt = cos t)

it becomes t*t dt

then it becomes t^2 dt

and the integral of this is t^3 / 3

Use a different variable or you'll really confuse yourself.
u = sin(t), du = cos(t)dt

So the integrand sin²(t) cos(t) dt becomes u² du