Question

Integral (7cos^3x+8sin^3x)÷3sin^2xcos^2x

Answer 1

SOLUTION

TO DETERMINE

$\displaystyle \int \sf{ \frac{7 { \cos}^{3} x + 8 { \sin}^{3}x }{3 { \sin}^{2}x { \cos}^{2} x } dx}$

EVALUATION

$\displaystyle \int \sf{ \frac{7 { \cos}^{3} x + 8 { \sin}^{3}x }{3 { \sin}^{2}x { \cos}^{2} x } dx}$

$= \displaystyle \int \sf{ \frac{7 { \cos}^{3} x }{3 { \sin}^{2}x { \cos}^{2} x } dx} + \int \sf{ \frac{ 8 { \sin}^{3}x }{3 { \sin}^{2}x { \cos}^{2} x } dx}$

$= \displaystyle \int \sf{ \frac{7 { \cos}^{} x }{3 { \sin}^{2}x } dx} + \int \sf{ \frac{ 8 { \sin}^{}x }{3 { \cos}^{2} x } dx}$

$= \displaystyle \frac{7}{3} \int \sf{ \csc x \cot x \: dx} + \frac{8}{3} \int \sf{ \sec x \tan x dx}$

$= \displaystyle - \frac{7}{3} \sf{ \csc x} + \frac{8}{3}\sf{ \sec x } + c$

Where C is integration constant

Note : csc x = cosec x

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