integrate 1/(x^2sqrt(1+x^2))
i know you have to use trig sub but iono how to cancel out some terms...much help is appreciated
Try -[sqrt(1+x^2)]/x
You will get two terms when you differentiate that. Arrange them with a common denominator and see what happens.
To integrate the given expression, you can indeed use a trigonometric substitution. Here's how you can proceed:
Step 1: Choose a suitable trigonometric substitution. In this case, let's substitute x = tan(theta), where theta is the angle in the interval (-pi/2, pi/2).
Step 2: Find the expression for dx in terms of d(theta) using the trigonometric identity dx = sec^2(theta) d(theta).
Step 3: Substitute x = tan(theta) and dx = sec^2(theta) d(theta) in the integral, which gives:
∫ 1/(x^2 * sqrt(1+x^2)) dx
= ∫ 1/((tan^2(theta)) * sqrt(1 + tan^2(theta))) * sec^2(theta) d(theta)
= ∫(sec^2(theta))/(tan^2(theta) * sec(theta)) d(theta)
Step 4: Simplify the expression:
Using the identity: sec^2(theta) = 1 + tan^2(theta), we can rewrite the integral as:
∫(1 + tan^2(theta))/(tan^2(theta) * sec(theta)) d(theta)
= ∫(1/tan^2(theta) + 1/(tan^2(theta) * sec(theta))) d(theta)
Step 5: Split the integral into two parts:
∫(1/tan^2(theta)) d(theta) + ∫(1/(tan^2(theta) * sec(theta))) d(theta)
Step 6: Simplify the first integral:
∫(1/tan^2(theta)) d(theta) = ∫cot^2(theta) d(theta)
Step 7: Use the trigonometric identity: cot^2(theta) = csc^2(theta) - 1, to rewrite the first integral:
∫cot^2(theta) d(theta) = ∫(csc^2(theta) - 1) d(theta)
= -∫d(theta) = -theta
Step 8: Simplify the second integral:
∫(1/(tan^2(theta) * sec(theta))) d(theta) = ∫(cos(theta)/sin^2(theta)) d(theta)
Step 9: Apply substitution: Let u = sin(theta), then du = cos(theta) d(theta), and rewrite the integral as:
∫(1/u^2) du = -1/u
Step 10: Substitute back for u:
= -1/sin(theta)
Step 11: Replace theta with its corresponding value in terms of x using the original substitution x = tan(theta). One way to do that is by using the right triangle definition of tangent:
tan(theta) = x
==> x = opposite/adjacent = sin(theta)/cos(theta)
==> sin(theta) = x * cos(theta)
Applying this substitution to -1/sin(theta):
= -1/(x * cos(theta))
Step 12: Simplify further:
= -1/(x * sqrt(1 - sin^2(theta)))
= -1/(x * sqrt(1 - x^2cos^2(theta))) [using sin^2(theta) = 1 - cos^2(theta)]
Step 13: Recall the original substitution x = tan(theta):
= -1/(x * sqrt(1 - x^2 * (1/(1 + x^2))^2))
= -1/(x * sqrt(1 - x^2/(1 + x^2)))
= -1/(x * sqrt((1 + x^2 - x^2)/(1 + x^2)))
= -1/(x * sqrt(1/(1 + x^2)))
= -sqrt(1 + x^2)/(x * sqrt(1 + x^2))
= -1/x
Hence, the definite integral of 1/(x^2 * sqrt(1 + x^2)) is -1/x + C, where C is the constant of integration.