Mathematics

Solve $$\displaystyle\int {\dfrac{{6{{\sin }^3}x + 5{{\cos }^3}x}}{{{{\sin }^2}x\,{{\cos }^2}x}}\,\,dx} $$


SOLUTION

Given that:


$$=\int {\dfrac{{6{{\sin }^3}x + 5{{\cos }^3}x}}{{{{\sin }^2}x\,{{\cos }^2}x}}\,\,dx} $$


$$=\int{\dfrac{6\,{\sin}x}{{\cos}^2{x}}}\,dx +\,\int{\dfrac{5\,{\cos}x}{{\sin}^2{x}}}\,dx$$


$$=\int6{\tan{x}\sec{x}}\, dx +\int5{\cot{x}\csc{x}}\,dx$$


$$\left[\because \int{\tan{x}\sec{x}}\,=\sec{x}, \int{\cot{x}\csc{x}}\,dx = -\csc{x}\right]$$


$$=6\sec{x}\,+5(\,-\csc{x})\, +constant$$


$$=6\sec{x}-5\csc{x}\,+constant$$

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Subjective Medium Published on 17th 09, 2020
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