Answer to Question #110775 in Calculus for Magano

Question

Find the indefinite integral of the following functions
Y= 3t^2+2e^3t+1/t+ cos 3t
Y=x/2+3x^2

Answer 1

$∫(3t^2+2e^{3t} +1/t+cos3t)dt=3∫t^2 dt+2∫e^{3t}dt +∫dt/t+∫cos(3t)dt=$ $=3⋅(1/3) t^3+2⋅(1/3) ⅇ^{3t}+ln⁡|t|+(1/3) sin⁡(3t)+C=t^3+(2/3) ⅇ^{3t}+ln⁡|t|+(1/3) sin(⁡3t)+C\\$

$\\2) ∫( x/2+3x^2)dx=(1/2) ∫xdx+3∫x^2 dx=$

$=(1/2)⋅(1/2) x^2+3⋅(1/3) x^3+C=(1/4) x^2+x^3+C$