[tex]integrate (\sin^{2}(x) \cos ^{2} (x) ) \div (sin^{5}(x) + \cos^{3}(x) \sin^{2}(x) + \sin^{3}(x) \cos ^{2}(x) + \c

Question

integrate (\sin^{2}(x) \cos ^{2} (x)  ) \div (sin^{5}(x) +  \cos^{3}(x)  \sin^{2}(x)  +  \sin^{3}(x) \cos ^{2}(x) +  \cos^{5}(x))^{2}

in progress 0
Nevaeh 5 months 2021-07-08T03:03:37+00:00 1 Answers 0 views 0

Answers ( )

    0
    2021-07-08T03:04:48+00:00

    EXPLANATION.

    \sf \implies \displaystyle \int \dfrac{sin^{2}x \ cos^{2}x  \ dx}{[sin^{5} x + cos^{3} x.sin^{2}x + sin^{3}x.cos^{2} x + cos^{5}x ]^{2}    }

    As we know that,

    We can write equation as,

    \sf \implies \displaystyle  \int \dfrac{sin^{2}x . cos^{2}x \ dx  }{[sin^{2}x (sin^{3} x + cos^{3} x )+ cos^{2}x (sin^{3}x + cos^{3} x)]^{2}    }

    \sf \implies \displaystyle  \int \dfrac{sin^{2}x. cos^{2}x \ dx }{[(sin^{2}x + cos^{2}x)  (sin^{3}x + cos^{3} x)]^{2}  }

    As we know that,

    Formula of :

    ⇒ sin²x + cos²x = 1.

    \sf \implies \displaystyle  \int\dfrac{sin^{2} x. cos^{2} x \ dx }{[(sin^{3}x + cos^{3} x)]^{2}  }

    \sf \implies \displaystyle  \int \dfrac{sin^{2}x . cos^{2}x \ dx  }{cos^{6}x \bigg(\dfrac{sin^{3} x}{cos^{3}x }\ + \dfrac{cos^{3}x }{cos^{3}x }  \bigg)^{2} }

    \sf \implies \displaystyle  \int\dfrac{sin^{2} x . cos^{2}x \ dx }{cos^{6}x (tan^{3} x + 1)^{2}  }

    \sf \implies \displaystyle  \int\dfrac{\bigg(\dfrac{sin^{2}x }{cos^{2}x } \ \times \dfrac{cos^{2}x }{cos^{4}x } \bigg)dx }{[tan^{3}x + 1]^{2}  }

    \sf \implies \displaystyle \int \dfrac{tan^{2} x . sec^{2}x \ dx }{[tan^{3}x + 1 ]^{2}  }

    As we know that,

    By using substitution method, we get.

    Let we assume that,

    ⇒ tan³x + 1 = t.

    Differentiate w.r.t x, we get.

    ⇒ 3tan²xsec²x dx = dt.

    ⇒ tan²xsec²x dx = dt/3.

    Substitute the value, we get.

    \sf \implies \displaystyle \dfrac{1}{3} \int \dfrac{dt}{t^{2} }

    \sf \implies \displaystyle -\bigg(\dfrac{1}{3} \times \dfrac{1}{t} \bigg)+ C.

    Put the value of t = tan³x + 1 in equation, we get.

    \sf \implies \displaystyle -\dfrac{1}{3(tan^{3} x + 1)} + C

                                                                                                                           

    MORE INFORMATION.

    (1) = If the integration is in the form.

    ∫eˣ [f(x) + f'(x)]dx = eˣ f(x) + c.

    (2) = If the integrals is in the form.

    ∫ [xf'(x) + f(x)]dx = x f(x) + c.

Leave an answer

Browse

9:3-3+1x3-4:2 = ? ( )