Question

Evaluate: $\int \frac{{\sin }^{6}x+{\cos }^{6}x}{{\sin }^{2}x\cdot {\cos }^{2}x}dx.$

Answer 1

Let $I=\int \frac{si{n}^{6}x+co{s}^{6}x}{si{n}^{2}xco{s}^{2}x}dx$

$=\int \frac{{\left(si{n}^{2}x\right)}^3+{\left(co{s}^{2}x\right)}^3}{si{n}^{2}xco{s}^{2}x}dx$

$=\int \frac{(si{n}^{2}x+co{s}^{2}x)(si{n}^{4}x+co{s}^{4}x-si{n}^{2}xco{s}^{2}x)}{si{n}^{2}xco{s}^{2}x}dx$

$=\int \frac{{(si{n}^{2}x+co{s}^{2}x)}^2-3si{n}^{2}xco{s}^{2}x}{si{n}^{2}xco{s}^{2}x}dx$

$=\int (\frac{1}{si{n}^{2}xco{s}^{2}x}-3)dx$

$=\int \frac{si{n}^2+co{s}^{2}x}{si{n}^{2}xco{s}^{2}x}dx-3x+C$

$=\int se{c}^{2}xdx+\int cose{c}^{2}xdx-3x+C$

$I=tanx-cotx-3x+C$