Question

Integration 2x/(x²+1)(x²+2)²

Answer 1

I = ∫2xdx/(x² + 1)(x² + 2)
Let x² = z
Differentiate with respect to x
2xdx = dz , put it in above Integration.
I = ∫dz/(z + 1)(z + 2)²
Now, you can see this can be solved by using partial fraction concepts
Let's try to do :
1/(z + 1)(z + 2)² = A/(z + 1) + B/(z + 2) + C/(z + 2)²
Multiply both sides, (z + 1)(z + 2)² ,
1 = A(z + 2)² + B(z + 1)(z + 2) + C(z +1)
= A(z² + 4z + 4) + B(z² + 3z + 2) + C(z + 1)
= (A + B)z² + (4A + 3B + C)z + (4A + 2B + C)
Compare both sides,
A + B = 0, 4A + 3B + C = 0 and (4A + 2B + C) = 1 solve it
A + C + 3(A + B) = 0, ⇒A + C = 0 ----(1)
Again, 2A + 2(A + B) + C = 1 ⇒2A + C = 1 ----(2)
Solve equations (1) and (2)
A = 1 and C = -1 , put A = 1 in A + B = 0 ⇒B = -A = -1

Hence,
I = ∫dz/(z + 1)(z + 2)² = ∫dz/(z + 1) + (-1)∫dz/(z + 2) + (-1)∫dz/(z + 2)²
= ln(z + 1) - ln(z + 2) + 1/(z + 2) + K
Now, put z = x²
I = ln(x² + 1) - ln(x² + 2) + 1/(x² + 2) + K, here k is constant.