Question

Evaluate the integral\(\int {\frac{{{{\rm{x}}^{\rm{2}}}}}{{{{\left( {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} \right)}^{{\rm{3/2}}}}}}} {\rm{dx}}\).

Short answer

The integral value of the given equation is\(\int {\frac{{{{\rm{x}}^{\rm{2}}}}}{{{{\left( {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} \right)}^{{\rm{3/2}}}}}}} {\rm{dx = }}\frac{{\rm{x}}}{{\sqrt {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} }}{\rm{ - si}}{{\rm{n}}^{{\rm{ - 1}}}}\left( {\frac{{\rm{x}}}{{\rm{2}}}} \right){\rm{ + c}}\).

Step-by-step solution

Expand the equation.

\({\rm{I = }}\int {\frac{{{{\rm{x}}^{\rm{2}}}}}{{{{\left( {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} \right)}^{{\rm{3/2}}}}}}} {\rm{dx}}\)

Let \({\rm{x = 2sint}} \to {\rm{dx = 2costdt}}\)

\(\begin{aligned}{c}{\rm{I = }}\int {\frac{{{\rm{4si}}{{\rm{n}}^{\rm{2}}}{\rm{t}}}}{{{{\left( {{\rm{4 - 4si}}{{\rm{n}}^{\rm{2}}}{\rm{t}}} \right)}^{{\rm{3/2}}}}}}} {\rm{ \times 2costdt}}\\{\rm{ = }}\int {\frac{{{\rm{4si}}{{\rm{n}}^{\rm{2}}}{\rm{t}}}}{{{{\left( {{\rm{4}}\left( {{\rm{1 - si}}{{\rm{n}}^{\rm{2}}}{\rm{t}}} \right)} \right)}^{{\rm{3/2}}}}}}} {\rm{ \times 2costdt}}\\{\rm{ = }}\int {\frac{{{\rm{4si}}{{\rm{n}}^{\rm{2}}}{\rm{t}}}}{{{\rm{8}}{{\left( {{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{t}}} \right)}^{{\rm{3/2}}}}}}} {\rm{ \times 2costdt}}\\{\rm{ = }}\int {\frac{{{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{t}}}}{{\left( {{\rm{co}}{{\rm{s}}^{\rm{3}}}{\rm{t}}} \right)}}} {\rm{ \times costdt}}\\{\rm{ = }}\int {\frac{{{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{t}}}}{{{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{t}}}}} {\rm{dt}}\\{\rm{ = }}\int {{\rm{ta}}{{\rm{n}}^{\rm{2}}}} {\rm{tdt}}\\{\rm{ = tant - t + C}}\end{aligned}\)

Evaluate the equation.

\(\begin{aligned}{c}{\rm{sint = }}\frac{{\rm{x}}}{{\rm{2}}} \to {\rm{t = si}}{{\rm{n}}^{{\rm{ - 1}}}}\left( {\frac{{\rm{x}}}{{\rm{2}}}} \right) \to {\rm{tant}}\\{\rm{ = }}\frac{{\rm{x}}}{{\sqrt {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} }}{\rm{I}}\\{\rm{ = }}\frac{{\rm{x}}}{{\sqrt {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} }}{\rm{ - si}}{{\rm{n}}^{{\rm{ - 1}}}}\left( {\frac{{\rm{x}}}{{\rm{2}}}} \right){\rm{ + c}}\\\int {\frac{{{{\rm{x}}^{\rm{2}}}}}{{{{\left( {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} \right)}^{{\rm{3/2}}}}}}} {\rm{dx = }}\frac{{\rm{x}}}{{\sqrt {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} }}{\rm{ - si}}{{\rm{n}}^{{\rm{ - 1}}}}\left( {\frac{{\rm{x}}}{{\rm{2}}}} \right){\rm{ + c}}\end{aligned}\)

Therefore, the integral value of the given equation is\(\int {\frac{{{{\rm{x}}^{\rm{2}}}}}{{{{\left( {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} \right)}^{{\rm{3/2}}}}}}} {\rm{dx = }}\frac{{\rm{x}}}{{\sqrt {{\rm{4 - }}{{\rm{x}}^{\rm{2}}}} }}{\rm{ - si}}{{\rm{n}}^{{\rm{ - 1}}}}\left( {\frac{{\rm{x}}}{{\rm{2}}}} \right){\rm{ + c}}\).