Question

Solve:
$dy over dx space equals space cos cubed straight x space sin to the power of 4 straight x plus straight x square root of 2 straight x plus 1 end root$

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Solution

The given differential equation is

dy over dx space equals space cos cubed straight x space sin to the power of 4 straight x plus straight x square root of 2 straight x plus 1 end root

Separating the variables, we get,

dy space equals space left parenthesis cos cubed straight x space sin to the power of 4 straight x plus straight x square root of 2 straight x plus 1 end root right parenthesis space dx

therefore space space space integral dy space equals space integral left parenthesis cos cubed straight x space sin to the power of 4 straight x plus straight x square root of 2 straight x plus 1 end root right parenthesis space dx

or space space space integral dy space equals space integral cos cubed straight x space sin to the power of 4 straight x space dx plus integral straight x space square root of 2 straight x plus 1 end root space dx

...(1)
Let I =

integral space cos cubed straight x space sin to the power of 4 straight x space dx space equals space integral cos squared straight x. space sin to the power of 4 straight x. space cosx space dx

equals space integral left parenthesis 1 minus sin squared straight x right parenthesis space sin to the power of 4 straight x. space cosx space dx

Put

sin space straight x space equals space straight t comma space space space therefore space space cos space straight x space dx space equals space dt

therefore space space space straight I space equals space integral left parenthesis 1 minus straight t squared right parenthesis space straight t to the power of 4 space dt space equals space integral left parenthesis straight t to the power of 4 minus straight t to the power of 6 right parenthesis space dt space equals space 1 fifth space straight t to the power of 5 space minus space 1 over 7 straight t to the power of 7 space equals 1 fifth sin to the power of 5 straight x minus 1 over 7 sin to the power of 7 straight x

Let

straight I subscript 1 space equals space integral straight x square root of 2 straight x plus 1 end root space dx

Put

square root of 2 straight x plus 1 end root space equals space straight y comma space space space space space therefore space space 2 straight x plus 1 space equals space straight y squared space space rightwards double arrow space space space 2 straight x space equals space straight y squared minus 1 space space rightwards double arrow space straight x space equals space 1 half left parenthesis straight y squared minus 1 right parenthesis

therefore space space space space dx space equals space 1 half space 2 straight y space dy space equals space straight y space dy therefore space space space straight I subscript 1 space equals space integral fraction numerator straight y squared minus 1 over denominator 2 end fraction. straight y. space straight y space dy space equals space 1 half integral left parenthesis straight y squared minus 1 right parenthesis space straight y squared space dy space equals space 1 half integral left parenthesis straight y to the power of 4 minus straight y squared right parenthesis space dy space equals 1 half open square brackets straight y to the power of 5 over 5 minus straight y cubed over 3 close square brackets space space space space space space space space space space space space equals space 1 over 10 straight y to the power of 5 space minus space 1 over 6 straight y cubed space equals space 1 over 10 left parenthesis 2 straight x plus 1 right parenthesis to the power of 5 divided by 2 end exponent minus space 1 over 6 left parenthesis 2 straight x plus 1 right parenthesis to the power of 3 divided by 2 end exponent

therefore space space space space from space left parenthesis 1 right parenthesis comma space we space get comma space space space space space space space space space space space integral dy space equals space 1 fifth sin to the power of 5 straight x minus 1 over 7 sin to the power of 7 straight x plus 1 over 10 left parenthesis 2 straight x plus 1 right parenthesis to the power of 5 divided by 2 end exponent minus 1 over 6 left parenthesis 2 straight x plus 1 right parenthesis to the power of 3 divided by 2 end exponent therefore space space space space straight y space equals space 1 fifth sin to the power of 5 straight x minus 1 over 7 sin to the power of 7 straight x plus 1 over 10 left parenthesis 2 straight x plus 1 right parenthesis to the power of 5 divided by 2 end exponent minus 1 over 6 left parenthesis 2 straight x plus 1 right parenthesis to the power of 3 divided by 2 end exponent plus straight c

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