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what is the integration of the following statement ∫(x^2) sqrt(9 + 4x^2) dx

what is the integration of the following statement
∫(x^2) sqrt(9 + 4x^2) dx

Grade:12th pass

1 Answers

Aditya Gupta
2081 Points
4 years ago
(x^2) sqrt(9 + 4x^2)= 3(x^2) sqrt(1 + (2x/3)^2)
now put y= 2x/3
so we will need to know how to solve (y^2) sqrt(1 + y^2) 
this can be solve using integration by parts taking first function as y and second fn as y*sqrt(1 + y^2) 
obviously the integral of y*sqrt(1 + y^2) can be found out by substituting z= 1+y^2
so at last we will have to find the inregral (1 + y^2)^3/2 dy which can be easily solved by substituting y= tan(u) so integral becomes sec^3u*sec^2udu= sec^5u du = cosu/cos^6u du= cosu/(1 – sin^2u)^3 du
now put v= sinu so we have to solve the integral dv/(1 – v^2)^3= dv/(1 – v)^3(1+v)^3 which can be easily solved by expansion using partial fractions. even wolfram alpha steps are this long :( please aprove :)

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