find intergral (x^2)/sqrt(x^2+25)
To find the integral of (x^2)/(sqrt(x^2+25)), we can use a trigonometric substitution. Let's follow these steps:
Step 1: Identify the trigonometric substitution. In this case, let's substitute x = 5 tan(theta). This substitution helps simplify the integral.
Step 2: Find dx in terms of d(theta). Since x = 5 tan(theta), we can differentiate both sides with respect to theta.
dx = 5 sec^2(theta) d(theta)
Step 3: Replace x^2 with its equivalent in terms of theta. Since x = 5 tan(theta),
x^2 = (5 tan(theta))^2 = 25 tan^2(theta)
Step 4: Represent sqrt(x^2+25) in terms of theta. Using the Pythagorean identity, we have:
sqrt(x^2+25) = sqrt((5 tan(theta))^2 + 25) = sqrt(25 tan^2(theta) + 25) = 5 sec(theta)
Step 5: Rewrite the integral with the substitution and its derived values:
∫(x^2)/(sqrt(x^2+25)) dx = ∫((25 tan^2(theta))/(5 sec(theta))) (5 sec^2(theta) d(theta))
= 5 ∫(tan^2(theta) sec(theta)) sec^2(theta) d(theta)
Step 6: Simplify the integral:
∫(tan^2(theta) sec(theta)) sec^2(theta) d(theta) = 5 ∫(tan^2(theta) sec^3(theta)) d(theta)
Step 7: Apply the reduction formula for integrals involving powers of sec(theta) to reduce the power of sec(theta):
∫(tan^2(theta) sec(theta)) sec^2(theta) d(theta) = 5 ∫(tan^2(theta) sec(theta)) d(theta) - 5 ∫(tan^2(theta)) d(theta)
Step 8: Evaluate the two integrals individually:
5 ∫(tan^2(theta) sec(theta)) d(theta) can be solved using integration by parts.
-5 ∫(tan^2(theta)) d(theta) can be represented as -5 ∫(sec^2(theta) - 1) d(theta)
Step 9: After integrating both parts, substitute x back in terms of theta to obtain the final result in terms of x.
Although the integration process can be lengthy and involves several substitutions and trigonometric identities, following these steps will lead you to the solution of the integral (∫) of (x^2)/(sqrt(x^2+25)).