The result of
∫2x(5−x)3dx=....
Explanation
If integration by parts ∫u⋅dv=u⋅v−∫vdu, then ∫2x(5−x)3dx
Let
Let
u=2x⇒du=2dx
dv=(5−x)3dx⇒v=∫dv=−41(5−x)4
Applying Integration by Parts Formula
Therefore
∫u⋅dv=2x(−41(5−x)4)−∫(−41(5−x)4)2dx
=(−2x(5−x)4)+21(−51(5−x)5)+C
=−2x(5−x)4−101(5−x)5+C
Simplifying
Factor out (5−x)4
=(2−x−10(5−x))(5−x)4+C
=(10−5x−(5−x))(5−x)4+C
=(10−5x−5+x)(5−x)4+C
=−101(4x+5)(5−x)4+C
Therefore, the result of ∫2x(5−x)3dx=−101(4x+5)(5−x)4+C