How do you evaluate $\int \frac{y^2}{y^2+d^2}dy$?

Question

$\color{green}{question}$:

How do you evaluate this integral?

$$\int \frac{y^2}{y^2+d^2}dy=y-d\,{\tan}^{-1}\left ( \frac{y}{d} \right )+\mathrm{constant}$$

$\color{green}{I~know}$ I should use the change of variables, But I do not know how to do.

Thank you for any hint.

@BabakS. thank you :) your comment give me incentive for more activities. — Software, Aug 24 '13 at 20:05
As a general rule, if you want to integrate the quotient of two polynomials where the degree of the numerator is at least as large as the degree of the denominator, then you want to divide first. — user84413, Aug 24 '13 at 21:14

score 6 · Answer 1 · answered Aug 24 '13 at 17:46

6

$$\frac{y^2}{y^2+d^2} = \frac{y^2 + d^2}{y^2+d^2} - \frac{d^2}{y^2+d^2}$$

The latter is an arctangent integral.

answered Aug 24 '13 at 17:46

Eric Auld

28,127

thank you, Excellent. $\Large\color{blue}{+}$ – Software Aug 24 '13 at 17:51

score 6 · Accepted Answer · answered Aug 24 '13 at 17:49

6

Hint

$$\frac{y^2}{y^2+d^2}=\frac{y^2+d^2-d^2}{y^2+d^2}=1-\frac{d^2}{y^2+d^2}$$
$$\frac{d^2}{y^2+d^2}=\frac{1}{(y/d)^2+1}=\frac{1}{t^2+1}$$
$$\int\frac{dt}{t^2+1}=\arctan t+C$$

answered Aug 24 '13 at 17:49

thanks for your hint $\Large\color{red}{+}$ – Software Aug 24 '13 at 17:55
You're welcome. – Aug 24 '13 at 17:56
1

I think this is the best answer so $\huge\color{green}{✓}$ for you :) thank you again. – Software Aug 24 '13 at 17:58

score 0 · Answer 3 · edited Aug 24 '13 at 19:34

0

Maple code with(Student[Calculus1]):IntTutor$(y^2/(y^2 +d^2),y$); produced the output which can be seen here. It coincides with the excellent answer by Eric Auld.

edited Aug 24 '13 at 19:34

Harish Kayarohanam

1,980
2
15
24

answered Aug 24 '13 at 19:12

user64494

5,811

thank you, This code is useful for me so +1 for you. :) – Software Aug 24 '13 at 19:27
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It would be preferred to insert the document linked as a image in your answer itself. – user642796 Sep 09 '13 at 08:35

How do you evaluate $\int \frac{y^2}{y^2+d^2}dy$?

3 Answers3