Effie's question via email about an indefinite integral.

[Prove It]

What is the indefinite integral (with respect to t) of $\displaystyle \begin{align*} 50\,t\cos{ \left( 5\,t^2 \right) } \end{align*}$?

$\displaystyle \begin{align*} \int{ 50\,t\cos{\left( 5\,t^2 \right) } \,\mathrm{d}t } &= 5\int{ 10\,t\cos{ \left( 5\,t^2 \right) }\,\mathrm{d}t } \end{align*}$

Let $\displaystyle \begin{align*} u = 5\,t^2 \implies \mathrm{d}u = 10\,t\,\mathrm{d}t \end{align*}$ and the integral becomes

$\displaystyle \begin{align*} 5\int{ 10\,t\cos{ \left( 5\,t^2 \right) } \,\mathrm{d}t } &= 5\int{ \cos{(u)}\,\mathrm{d}u } \\ &= 5\sin{(u)} + C \\ &= 5\sin{ \left( 5\,t^2 \right) } + C \end{align*}$

 

[Mark44]

And the easy check is to differentiate the result to see if you get back the original antiderivative.