What substitution could I use to integrate
a/(a^2 + x^2)^3/2 dx
Let u = x/(x^2 + a^2)^1/2
and you will find that
(1/a^2)* du
= integral of dx/(x^2+a^2)^3/2
which is the integral you want.
Therefore u/a^2
= (x/a^2)/(x^2 + a^2)^1/2
is the answer.
Computer program says the answer is
x/(a*(a^2 + x^2)^(1/2))
which is slightly different from your answer. Thanks so much for the help on this one. I was really stuck.
I did x = a tan u
dx = a (sec u)^2 du
int of a/(a^2 + x^2)^3/2 dx
= int of (a sec u)^2/(a^2 + (a tan u)^2)^3/2 du
= int of (a sec u)^2/(a sec u)^3 du
= int of (cos u)/a du
= (sin u)/a + K
since u = atan (x/a)
= x/(a*(a^2 + x^2)^(1/2)) + K
Thanks again...
To integrate the expression ∫a/(a^2 + x^2)^(3/2) dx, a substitution called "trigonometric substitution" can be used.
1. Start by making the substitution x = a * tan(u).
2. Find dx by taking the derivative of both sides with respect to u: dx = a * sec^2(u) du.
3. Substitute the values of x and dx in the integral expression to get ∫a/(a^2 + (a * tan(u))^2)^(3/2) * a * sec^2(u) du.
4. Simplify the expression by canceling out the common factors of "a" and "sec^2(u)". It becomes ∫1/(a^2 * (1 + tan^2(u)))^(3/2) du.
5. Use the trigonometric identity tan^2(u) + 1 = sec^2(u) to rewrite the expression as ∫1/(a^2 * sec^2(u))^(3/2) du.
6. Simplify further to get ∫1/(a^2 * sec^2(u))^(3/2) du = ∫1/(a^3 * sec^3(u)) du.
7. Now, the integral can be simplified to ∫cos^3(u)/a^3 du by using the identity sec(u) = 1/cos(u).
8. Finally, integrate the expression with respect to u using standard integration techniques or by using the reduction formula for powers of cosine.
Note: Trigonometric substitution is often used for integrals involving expressions of the form (a^2 + x^2)^(m/n), where m and n are integers.