Answer to Question #101489 in Calculus for Mal

Question #101489
1. Integrate with the placement method: ∫cosxsin2xdx

2. Integrate using the partial integration method
a.) ∫e^x*cosx dx
b.) ∫x^2*e^3x dx
1
Expert's answer
2020-01-20T09:45:12-0500

Solution.

  1. Using the formula
"sin2x=2sinxcosx"

get


"\\intop cosxsin2xdx=\\intop 2(cosx)^2sinxdx"

We use a replacement k=cosx, therefore dk=-sinxdx. As result making a replacement, we get

"\\intop 2(cosx)^2sinxdx=-\\intop 2k^2dk=-\\frac {2k^3}{3}+C"

where C is constant. Returning to the replacement, we get


"\\intop cosxsin2xdx=-\\frac{2(cosx)^3}{3} +C"


2. Integrate using the partial integration method

a)


"\\begin{vmatrix}\n u=e^x & dv=cosxdx \\\\\n du=e^xdx & v=sinx\n\\end{vmatrix}"

Therefore we write the integral as


"\\intop e^xcosxdx=e^xsinx-\\intop e^xsinxdx"

Using the partial integration method get


"\\begin{vmatrix}\n u=e^x & dv=sinxdx \\\\\n du=e^xdx & v=-cosx\n\\end{vmatrix}"

"e^xsinx-\\intop e^xsinxdx=e^xsinx+e^xcosx-\\intop e^xcosx"

Hence


"\\intop e^xcosxdx=\\frac{e^xsinx+e^xcosx}{2}+C"

where C is constant.

b. Using the partial integration method get


"\\begin{vmatrix}\n u=x^2 & dv=e^{3x}dx \\\\\n du=2xdx & v=\\frac{e^{3x}}{3}\n\\end{vmatrix}"

Therefore we write the integral as


"\\intop x^2e^{3x}dx=\\frac{x^2e^{3x}}{3} -\\frac{2}{3}\\intop xe^{3x}dx"

Using the partial integration method get


"\\begin{vmatrix}\n u=x & dv=e^{3x}dx \\\\\n du=dx & v=\\frac{e^{3x}}{3}\n\\end{vmatrix}"

Therefore we write the integral as


"\\intop x^2e^{3x}dx=\\frac{x^2e^{3x}}{3} -\\frac{2xe^{3x}}{9} +\\frac{2}{9}\\intop e^{3x}dx"

As result


"\\intop x^2e^{3x}dx=\\frac{x^2e^{3x}}{3} -\\frac{2xe^{3x}}{9} +\\frac{2e^{3x}}{27}+C"

where C is constant.

Answer. 1.


"\\intop cosxsin2xdx=-\\frac{2(cosx)^3}{3} +C"

where C is constant.

2 a.


"\\intop e^xcosxdx=\\frac{e^xsinx+e^xcosx}{2}+C"

where C is constant.

2 b.


"\\intop x^2e^{3x}dx=\\frac{x^2e^{3x}}{3} -\\frac{2xe^{3x}}{9} +\\frac{2e^{3x}}{27}+C"

where C is constant.


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