Question

Integration Procedures (Integration by Parts)

∫ x³ cos 5x dx

Answer 1

Part 1

dv=cos(5x)dx

u=x³

du=3x²dx

v=1/5 sin(5x)

$\smallint{udv}=uv-\smallint{vdu}$

$\smallint{{x^3}cos(5x)}dx=1/5x^3sin(5x)dx-3/5\int{x^2sin(5x)}dx$

Part 2

dv=sin(5x)dx

u=x²

du=2xdx

v=-1/5 cos(5x)

$\smallint{{x^2}sin(5x)}dx=-1/5x^2cos(5x)+2/5\int{xcos(5x)}dx$

Part 3

dv=cos(5x)dx

u=x

du=dx

v=1/5 sin(5x)

$\smallint{{x}cos(5x)}dx=-1/5xsin(5x)-1/5\int{sin(5x)}dx=1/5xsin(5x)+1/25cos(5x)+C$

Total

$\smallint{{x^3}cos(5x)}dx=1/5x^3sin(5x)-3/5(-1/5x^2cos(5x)+2/5(1/5xsin(5x)+1/25cos(5x)+C))=1/5x^3sin(5x)+3/25x^2cos(5x)-6/125xsin(5x)-12/625cos(5x)+C$