[integrals]2/tsqrt(t^4+25)
integrals of two over t square root of t to the 4th plus 25
To find the integral of 2/(t√(t^4+25)), we can use a trigonometric substitution. Let u = t^2, which means du = 2t dt. Rearranging, we get dt = du/(2t).
Now, we need to express the expression inside the square root using u instead of t. Substituting u = t^2 into t√(t^4+25), we get (√(u^2+25)). Notice that we now have only u in the expression, which is useful for applying trigonometric substitution.
Next, we need to simplify the expression for dt. Substituting dt = du/(2t) and using u = t^2, we get dt = du/(2√u).
Now let's combine everything into the integral:
∫(2/(t√(t^4+25))) dt = ∫(2/(t(√(u^2+25))) du/(2√u)
= ∫(1/(t√(u^2+25))) du/√u
Simplifying further, we have:
∫(1/(√(u^2+25))) du/u
We can solve this integral using trigonometric substitution. Let u = 5tan(theta), which means du = 5sec^2(theta) d(theta). Rearranging, we have d(theta) = du/(5sec^2(theta)).
Now, let's express the expression inside the integral using theta instead of u. Substituting u = 5tan(theta), we get (√(25tan^2(theta)+25)), which simplifies to 5sec(theta).
Next, we need to simplify the expression for d(theta). Substituting d(theta) = du/(5sec^2(theta)) and using u = 5tan(theta), we get d(theta) = du/5.
Now, let's rewrite the integral using theta:
∫(1/(√(u^2+25))) du/u = ∫(1/(√(25tan^2(theta)+25))) (du/5)
Simplifying further, we have:
(1/5) ∫(1/(√(tan^2(theta)+1))) du
Now, we can simplify the expression inside the integral using the trigonometric identity: sec^2(theta) = tan^2(theta) + 1. Thus, we can replace the expression inside the square root with sec(theta).
(1/5) ∫(1/(√(sec^2(theta)))) d(theta)
(1/5) ∫(1/sec(theta)) d(theta)
(1/5) ∫cos(theta) d(theta)
Finally, we integrate ∫cos(theta) d(theta) to get sin(theta):
(1/5) ∫cos(theta) d(theta) = (1/5)sin(theta)
Replacing theta with its original variable, t, we have:
∫(2/(t√(t^4+25))) dt = (1/5)sin(theta)
So, the integral of 2/(t√(t^4+25)) with respect to t is (1/5)sin(theta) + C, where theta is determined by the relation u = 5tan(theta) and C is the constant of integration.