Question

Evaluate the integral\(\int\limits_{\rm{0}}^{{{\rm{\pi }} \mathord{\left/

{\vphantom {{\rm{\pi }} {\rm{2}}}} \right.

\kern-\nulldelimiterspace} {\rm{2}}}} {{\rm{co}}{{\rm{s}}^{\rm{3}}}{\rm{xsin2xdx}}} \).

Short answer

The integral value of the given equation is\(\int\limits_{\rm{0}}^{{{\rm{\pi }} \mathord{\left/

{\vphantom {{\rm{\pi }} {\rm{2}}}} \right.

\kern-\nulldelimiterspace} {\rm{2}}}} {{\rm{co}}{{\rm{s}}^{\rm{3}}}{\rm{xsin2xdx}}} {\rm{ = }}\frac{{\rm{2}}}{{\rm{5}}}\).

Step-by-step solution

Expand the equation.

\(\int\limits_{\rm{0}}^{{{\rm{\pi }} \mathord{\left/

{\vphantom {{\rm{\pi }} {\rm{2}}}} \right.

\kern-\nulldelimiterspace} {\rm{2}}}} {{\rm{co}}{{\rm{s}}^{\rm{3}}}{\rm{xsin2xdx}}} \)

Known value \({\rm{sin2x = 2sinx \times cosx}}\)

\({\rm{ = }}\int_{\rm{0}}^{{{\rm{\pi }} \mathord{\left/

{\vphantom {{\rm{\pi }} {\rm{2}}}} \right.

\kern-\nulldelimiterspace} {\rm{2}}}} {{\rm{co}}{{\rm{s}}^{\rm{3}}}{\rm{x(2sinx \times cosx)dx}}} \)

\({\rm{ = 2}}\int_{\rm{0}}^{{{\rm{\pi }} \mathord{\left/

{\vphantom {{\rm{\pi }} {\rm{2}}}} \right.

\kern-\nulldelimiterspace} {\rm{2}}}} {{\rm{co}}{{\rm{s}}^{\rm{4}}}{\rm{x(sinxdx)}}} \)

\({\rm{ = - 2}}\int_{\rm{0}}^{{{\rm{\pi }} \mathord{\left/

{\vphantom {{\rm{\pi }} {\rm{2}}}} \right.

\kern-\nulldelimiterspace} {\rm{2}}}} {{{{\rm{(cosx)}}}^{\rm{4}}}{\rm{( - sinxdx)}}} \)

Evaluate the equation.

Substitute \({\rm{cosx = u}}\)and\({\rm{ - sinxdx = du}}\). Integration's limits will shift over time. \(\int_{\rm{0}}^{{{\rm{\pi }} \mathord{\left/

{\vphantom {{\rm{\pi }} {\rm{2}}}} \right.

\kern-\nulldelimiterspace} {\rm{2}}}} {{\rm{to}}\int_{{\rm{cos0}}}^{{\rm{cos}}{{\rm{\pi }} \mathord{\left/

{\vphantom {{\rm{\pi }} {\rm{2}}}} \right.

\kern-\nulldelimiterspace} {\rm{2}}}} {{\rm{ = }}\int_{\rm{1}}^{\rm{0}} {} } } \)

\({\rm{ = - 2}}\int_{\rm{1}}^{\rm{0}} {{{\rm{u}}^{\rm{4}}}{\rm{du}}} \)

\({\rm{ = - 2}}\left( {\frac{{{{\rm{u}}^{\rm{5}}}}}{{\rm{5}}}} \right)_{\rm{1}}^{\rm{0}}\)

\({\rm{ = - 2}}\left( {\frac{{\rm{0}}}{{\rm{5}}}{\rm{ - }}\frac{{\rm{1}}}{{\rm{5}}}} \right)\)

\({\rm{ = }}\frac{{\rm{2}}}{{\rm{5}}}\)

Therefore, the integral value of the given equation is\(\int\limits_{\rm{0}}^{{{\rm{\pi }} \mathord{\left/

{\vphantom {{\rm{\pi }} {\rm{2}}}} \right.

\kern-\nulldelimiterspace} {\rm{2}}}} {{\rm{co}}{{\rm{s}}^{\rm{3}}}{\rm{xsin2xdx}}} {\rm{ = }}\frac{{\rm{2}}}{{\rm{5}}}\).