If ( 1+x^2 +x )¹⁰ ( 1+x^2 -x )^9 d x= 1 m ( ( 1+x^2 +x )^n (n 1+x^2 -x ) )+C where C is the constant of integration and m, n N , then m + n is equal to

Step-by-step solution

rationalise aligned & ( 1+x^2 +x )¹⁰ ( 1+x^2 -x )^9 ( 1+x^2 +x )^9 ( 1+x^2 +x )^9 d x & ( 1+x^2 +x )¹⁹ 1 d x aligned Put 1+ x ^2 + x = t ( x 1+ x ^2 +1 ) dx = dt dx = dt t 1+ x ^2 Now as 1+ x ^2 + x = t aligned & so 1+ x ^2 - x = 1 t & 1+ x ^2 = 1 2 ( t + 1 t ) aligned Thus I= t ¹⁹ dt t 1 2 ( t + 1 t ) aligned & 1 2 ( t ¹⁹+ t ¹⁷ ) dt & = 1 2 ( t ²⁰ 20 + t ¹⁸ 18 )+ C & = t ¹⁹ 360 [9 t + 10 t ]+ C & = t ¹⁹ 360 [9 ( t + 1 t )+ 1 t ]+ C & ( 1+ x ^2 + x )¹⁹ 360 [9 (2 1+ x ^2 )+ ( 1+ x ^2 - x ) ]+ C & ( 1+ x ^2 + x )¹⁹ 3