Integral of a vector valued function

[Lucid Dreamer]

I came across this integral of a vector valued function.
$$\int \mathbf A(t) \vec{w(t)} dt = \int \mathbf B(t)$$.
I want to isolate $\vec{w(t)}$ and so I multiply by $\left (\int \mathbf A(t) dt \right)^{-1}$ on both sides.
$$\left (\int \mathbf A(t) dt \right)^{-1} \int \mathbf A(t) \vec{w(t)} dt = \left (\int \mathbf A(t) dt\right)^{-1} \int \mathbf B(t) dt$$

I thought the correct form would be
$$\int \vec{w(t)} dt = \left (\int \mathbf A(t) dt\right)^{-1} \int \mathbf B(t) dt$$.

But it turns out I get the right answer if I take
$$\vec{w(t)} = \left (\int \mathbf A(t) dt \right)^{-1} \int \mathbf B(t) dt$$.

Can anyone show why the second form is correct?

 

[fresh_42]

Both formulas are wrong. Check it with $\omega(t)=t^2$ and $A(t)=e^t$. All we have is the integration by parts: $\displaystyle{\int } A(t)\omega(t)dt = \left(\int A(t)dt\right) \omega(t) - \int \left(\int A(t)dt\right)\omega'(t)dt$