Answer to Question #178349 in Calculus for Phyroe

Question #178349

Integration Procedures (Integration by Parts)


∫ x³ cos 5x dx


1
Expert's answer
2021-05-07T09:01:58-0400

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"



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