Why am i getting two different answers for two different ways of solving the integral.

Question

So the Integral is $$\int \frac {dt}{t(t+1)^2}$$ So i thought of it two ways.

1.Substituting $\displaystyle y=\frac 1t$

Solution: $\displaystyle \ln\left|\frac{t}{1+t}\right|-\frac{t}{1+t}$

2.Substituting $\displaystyle y=\frac {1}{1+t}$

Solution: $\displaystyle \ln\left|\frac{t}{1+t}\right|+\frac {1}{1+t}$

Here's the photo for reference enter image description here

Your answers differ by the constant $1$. So, both are valid. Add $1$ to your first answer and see. You'll get the second answer. — Souparna, Apr 17 '23 at 12:50
When in doubt that the integral is wrong, compute the derivative — Andrea Mori, Apr 17 '23 at 12:55
Does this answer your question? Getting different answers when integrating using different techniques — Martin R, Apr 17 '23 at 14:13

score 3 · Accepted Answer · answered Apr 17 '23 at 13:51

1.Substituting $\displaystyle y=\frac 1t$

Solution: $\displaystyle \ln\left|\frac{t}{1+t}\right|-\frac{t}{1+t}$

2.Substituting $\displaystyle y=\frac {1}{1+t}$

Solution: $\displaystyle \ln\left|\frac{t}{1+t}\right|+\frac {1}{1+t}$

Note that:

$$\frac{1}{1+t}=-\frac{t}{1+t}+1$$

your two solutions differ by a constant $1$, hence both of them are correct.

Why am i getting two different answers for two different ways of solving the integral.

1 Answers1