Use the reduction formulas in a table of integrals to evaluate the following integrals. ∫p2e−3pdp

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Use the reduction formulas in a table of integrals to evaluate the following integrals.  ∫p2e−3pdp

grbavit · Accepted Answer

Reduction formula for Exponential function is given as:  ∫xaebxdx=[1bxaebx]−[ab∫xa−1ebc]dx  we have to evaluate:  ∫p2e−3pdp  Using reduction formula, we will integrate it.  ∫p2e−3pdp=[1−3p2e−3p]−[2−3∫p2−1⋅e−3p]dp  =13p2e−3p+23∫pe−3pdp  Now, we will integrate∫pe−3pdp using integration by parts  ∫pe−3pdp=p∫e−3pdp−∫[ddx(p)∫e−3pdp]dp  =p⋅e−3p−3−∫[1⋅e−3p−3]dp  =−pe−3p3+13∫e−3pdp  =−pe−3p3+13⋅e−3p−3+c  =−pe−3p3−19e−3p+c  Putting above integration value in equation, we get   ∫p2e−3pdp=−13p2e−3p+23[−pe−3p3−19e−3p]+c  =−13p2e−3p−29pe−3p−227e−3p+c  =−e−3p(13p2+29+227)+c  Hence, this is required integration.

Use the reduction formulas in a table of integrals to evaluate the following integrals. int p^2e^{-3p}dp

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