Evaluate the given integral. ∫x2e−xdx

Question

Evaluate the given integral.  ∫x2e−xdx

sonSnubsreose6v · Accepted Answer

Step 1  To evaluate the integral:  ∫x2e−xdx  Formula used:  Integration by parts: ∫uvdx=u∫vdx−∫[dudx⋅∫vdx]dx  Step 2  Let u=x2 and v=e−x  Therefore,  ∫x2e−xdx=x2∫e−xdx−∫[dx2dx⋅∫e−xdx]dx  Apply the Power Rule: ddx(xa)=a⋅xa−1  Use the common integral: ∫eudu=eu  ∫x2e−xdx=−x2e−x−∫−2×⋅e−xdx  =−x2e−x+2∫x⋅e−xdx  Again apply the integration by parts to integrate ∫x⋅e−xdx:  ∫x⋅e−xdx=x∫e−xdx−∫[dxdx⋅∫e−xdx]dx  =−xe−x−∫−e−xdx  =−xe−x−e−x  Therefore,  ∫x2e−xdx=−x2e−x−∫−2x⋅e−xdx  =−x2e−x+2∫x⋅e−xdx

Cleveland Walters · Accepted Answer

∫x2e−xdx  Integration formula by parts:  ∫U(x)⋅dV(x)=U(x)⋅V(x)−∫V(x)⋅dU(x)  Put  U=x2  dV=e−xdx  Then:  dU=2xdx  V=−e−x  Therefore:  ∫x2⋅e−xdx=−x2e−x−∫−2x⋅e−xdx=−x2⋅e−x+∫2x⋅e−xdx  Find the integral  ∫2x⋅e−xdx  U=x  dU=dx  dV=2e−xdx  V=−2e−x  ∫2⋅x⋅e−xdx=−2⋅x⋅e−x−∫−2⋅e−xdx=−2x⋅e−x+∫2⋅e−xdx  Find the integral  ∫2⋅e−xdx=−2⋅e−x  Answer:  ∫x2⋅e−x=−x2⋅e−x−2x⋅e−x−2⋅e−x+C

Evaluate the given integral. \int x^{2}e^{-x}dx

Answered question

Answer & Explanation

New Questions in Calculus 2