Using trigonometric substitution. Do not use the following antiderivative calculator or other machine assistance. ∫(x^2 + 25) dx The substitution starts with the substitution of the trigonometric tangent (tanθ) opposite, which appears in a right triangle, because it is adjacent to the angle θ. The substitution is: x = 5tan(θ) Use the substitution equation to find the leg lengths of an associated right triangle. Then use the right triangle to find sec(θ). Write in terms of x. sec(θ) Take the derivative of both sides of the substitution equation to find dx in terms of dθ. dx = 5sec^2(θ) dθ Transform the original antiderivative into an antiderivative written in terms of θ, by using the results of (θ) and (0). Do not compute the transformed antiderivative. ∫(√(x^2 + 25)) dx =
The Correct Answer and Explanation is:
Here are the solutions to the problem:
a. sec(θ) = (√(x² + 25)) / 5
b. dx = 5sec²(θ) dθ
c. ∫√(x² + 25) dx = ∫ 25sec³(θ) dθ
Explanation
This problem demonstrates the process of transforming an integral using trigonometric substitution. The goal is to replace the variable ‘x’ with a trigonometric function of a new variable ‘θ’ to simplify the integrand, particularly the radical expression.
a. Finding sec(θ) in terms of x
The problem provides the substitution x = 5tan(θ). This substitution is chosen because the integral contains the form √(x² + a²), where a = 5.
- Relate to a Right Triangle: We can rearrange the substitution equation to get tan(θ) = x/5. In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side (SOH CAH TOA). Therefore, we can construct a right triangle where the side opposite angle θ is ‘x’ and the side adjacent to angle θ is ‘5’.
- Find the Hypotenuse: Using the Pythagorean theorem (a² + b² = c²), we can find the length of the hypotenuse:
(Adjacent)² + (Opposite)² = (Hypotenuse)²
5² + x² = (Hypotenuse)²
Hypotenuse = √(x² + 25) - Calculate sec(θ): The secant of an angle is the ratio of the hypotenuse to the adjacent side. Using the values from our triangle:
sec(θ) = Hypotenuse / Adjacent = (√(x² + 25)) / 5
b. Finding dx in terms of θ
To completely transform the integral, we must also substitute for ‘dx’. This is done by differentiating the substitution equation x = 5tan(θ) with respect to θ.
- Differentiate: We find the derivative of both sides:
d/dθ(x) = d/dθ(5tan(θ))
dx/dθ = 5 * sec²(θ) (since the derivative of tan(θ) is sec²(θ)) - Isolate dx: By treating dx/dθ as a fraction, we can multiply both sides by dθ to solve for dx:
dx = 5sec²(θ) dθ
c. Transforming the Integral
Now we use the results from parts (a) and (b) to rewrite the original integral, ∫√(x² + 25) dx, entirely in terms of θ.
- Substitute the Radical: From part (a), we found that √(x² + 25) is the hypotenuse. We also know that 5sec(θ) = √(x² + 25). So, we replace √(x² + 25) with 5sec(θ).
- Substitute dx: From part (b), we replace dx with 5sec²(θ) dθ.
- Combine and Simplify: Substitute these into the integral:
∫√(x² + 25) dx = ∫(5sec(θ)) * (5sec²(θ) dθ)Multiplying the terms together gives the final transformed integral:
∫ 25sec³(θ) dθ
