Solve ∫(x+2)/(2x^2+1)^3 dx
Since (x+2)/(2x2+1)3 can't be easily simplified into a more straightforward integral, we have to rely on some more advanced integration methods. Before trying things like substitution, or integration by parts, however, we'll check whether this can be solved using the reverse chain rule, as this often gets us to a simpler (and easier) solution.Before jumping into trying to solve it, remember that only certain types of integrals can be solved using the reverse chain rule. The two conditions are that:• The integral must be made up of two functions, either as a fraction (one divided by the other) or as a product (one times the other)• If you differentiate the inside of the brackets in the denominator, you get a function that is roughly the same as the numerator. What this means is that the coefficients and the numbers by themselves can change, but any important bits – powers of x, trig functions etc – have to be the same. For instance, 2sin(x) and 3sin(2x) are roughly the same, but 2sin(x) and sin(x2 ) are not.In this case we have a fraction, so it meets the first condition and is good so far. So let's check the second condition. In our case, differentiating 2x2 +1 gives 4x, which is roughly the same as x+2, since the powers of x are the same, so we can use the reverse chain rule. We first rewrite the integral as ∫(x+1)(2x2+1)-3 dx. Now to use the reverse chain rule, we simply write this line out again, increasing the power of the denominator by one. Then we divide by both the new power, and the differentiated denominator we found just now (4x). Therefore, ∫(x+1)(2x2+1)-3 dx = (x+1)(2x2+1)-2 / (-2)(4x) + CDon't forget the plus C!
