Question

integration of x^5/root 1+x^3 dx​

Answer 1

Answer:

Step-by-step explanation:

∫  x^5/root 1+x^3 dx​

  take x^5/2 common in root of denominator

∫x^5/root x^5/2(1/x^5/2+x^1/2 )

∫x^5/x^5 root ( x^-5/2 +x^1/2 )

cancel out x^5 and x^5

∫dx/root x^-5/2 +x^1/2

lets convert denominator in the form of root x^2 + a^2

x^-5/2 +x^1/2 - 2x^-1 + 2x^-1

(x^-5/4-x^1/4)^2 + ((root2) x^-1/2)^2

∫dx/root ((x^-5/4-x^1/4)^2 + ((root2) x^-1/2)^2

we know ∫dx/root(x^2+a^2) is log (x+ root x^+a^2 )

∫dx/ root(x^-5/4-x^1/4)^2 + ((root2) x^-1/2)^2

=log (x^-5/4-x^1/4 + root(x^-5/4-x^1/4)^2 + ((root2) x^-1/2)^2 ) /

-5/4 x^-9/4 -1/4x^-5/4