Question

How do you integrate `h(t)=(t+1)/(t^2+2t+2)` using the quotient rule?

Answer 1

Please see the explanation for the steps leading to the answer:

`int(t + 1)/(t^2 + 2t + 2)dx = (1/2)ln(t^2 + 2t + 2) + C`

Explanation:

The quotient rule does not pertain to integrals; it pertains to derivatives.

This integral is best integrated by u substitution:

let `u = t^2 + 2t + 2`, then `du = (2t + 2)dx`, therefore, we shall substitute `(1/2)du` for `(t + 1)dx`:

`int(t + 1)/(t^2 + 2t + 2)dx = (1/2)int(du)/u`

`int(t + 1)/(t^2 + 2t + 2)dx = (1/2)ln|u| + C`

Reverse the substitution for u:

`int(t + 1)/(t^2 + 2t + 2)dx = (1/2)ln|t^2 + 2t + 2| + C`

Because `t^2 + 2t + 2` can never be less than or equal to zero, we may drop the absolute value:

`int(t + 1)/(t^2 + 2t + 2)dx = (1/2)ln(t^2 + 2t + 2) + C`