7.1 Integration By Parts/4: Difference between revisions

Latest revision as of 03:02, 29 November 2022

$\int (xe^{-x})dx$

$u=x\qquad dv=e^{-x}$

$du=dx\qquad v=-e^{-x}$

$\int (xe^{-x})dx=x(-e^{-x})-\int (-e^{-x})=-xe^{-x}-\int (-e^{-x})dx$

$=-xe^{-x}+\int (e^{-x})dx=-xe^{-x}-e^{-x}+C$

$=-e^{-x}(x+1)+C$

@@ Line 1: / Line 1: @@
-<math>\int xe
+<math>\int (xe^{-x}) dx </math>
-</math>
+<math>u = x \qquad dv = e^{-x}</math> <br><br>
+<math>du = dx \qquad v = -e^{-x}</math>
+<math>\int (xe^{-x}) dx = x(-e^{-x}) - \int (-e^{-x}) = -xe^{-x}-\int (-e^{-x})dx</math>
+<math>=-xe^{-x}+\int (e^{-x})dx = -xe^{-x}-e^{-x}+C</math> <br><br>
+<math>=-e^{-x}(x+1)+C</math>