Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

∫242xx2−4dx\int_{2}^{4}\frac{2}{x\sqrt{x^{2}-4}}dx

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Calculusright chevron icon
Find indefinite integrals using the substitution and by partsright chevron icon

Full solution

Q. ∫242xx2−4dx\int_{2}^{4}\frac{2}{x\sqrt{x^{2}-4}}dx
  1. Substitution and Simplification: Substitute x=2sec⁡(θ)x = 2\sec(\theta) into the integral and simplify.∫2xx2−4dx=∫22sec⁡(θ)4sec⁡2(θ)−4⋅2sec⁡(θ)tan⁡(θ)dθ\int\frac{2}{x\sqrt{x^2-4}}dx = \int\frac{2}{2\sec(\theta)\sqrt{4\sec^2(\theta)-4}} \cdot 2\sec(\theta)\tan(\theta)d\theta =∫22sec⁡(θ)4(tan⁡2(θ)+1)−4⋅2sec⁡(θ)tan⁡(θ)dθ= \int\frac{2}{2\sec(\theta)\sqrt{4(\tan^2(\theta)+1)-4}} \cdot 2\sec(\theta)\tan(\theta)d\theta =∫22sec⁡(θ)4tan⁡2(θ)⋅2sec⁡(θ)tan⁡(θ)dθ= \int\frac{2}{2\sec(\theta)\sqrt{4\tan^2(\theta)}} \cdot 2\sec(\theta)\tan(\theta)d\theta =∫22sec⁡(θ)⋅2tan⁡(θ)⋅2sec⁡(θ)tan⁡(θ)dθ= \int\frac{2}{2\sec(\theta) \cdot 2\tan(\theta)} \cdot 2\sec(\theta)\tan(\theta)d\theta =∫24tan⁡(θ)⋅2sec⁡(θ)tan⁡(θ)dθ= \int\frac{2}{4\tan(\theta)} \cdot 2\sec(\theta)\tan(\theta)d\theta =∫(12)dθ= \int\left(\frac{1}{2}\right)d\theta
  2. Integration and Simplification: Integrate (12)dθ(\frac{1}{2})d\theta from 00 to π3\frac{\pi}{3}.
    ∫(12)dθ=(12)θ\int(\frac{1}{2})d\theta = (\frac{1}{2})\theta
    Evaluating from 00 to π3\frac{\pi}{3} gives us:
    (12)θ∣(\frac{1}{2})\theta | from 00 to π3\frac{\pi}{3} = (12)(π3)−(12)(0)=π6(\frac{1}{2})(\frac{\pi}{3}) - (\frac{1}{2})(0) = \frac{\pi}{6}
  3. Evaluation and Conversion: Convert the result back to xx. Since we made the substitution x=2sec⁡(θ)x = 2\sec(\theta), we need to find the corresponding θ\theta values for x=2x = 2 and x=4x = 4. However, we already have these values from the substitution step: θ=0\theta = 0 when x=2x = 2 and θ=π3\theta = \frac{\pi}{3} when x=4x = 4. Therefore, the definite integral from 22 to x=2sec⁡(θ)x = 2\sec(\theta)00 of the function x=2sec⁡(θ)x = 2\sec(\theta)11 with respect to xx is x=2sec⁡(θ)x = 2\sec(\theta)33.

More problems from Find indefinite integrals using the substitution and by parts

Question
Evaluate the integral.\newline∫−x4−3xdx \int-x 4^{-3 x} d x \newlineAnswer:
Get tutor helpright-arrow

Posted 2 months ago

Question
Evaluate the integral.\newline∫6x3e−2xdx \int 6 x^{3} e^{-2 x} d x \newlineAnswer:
Get tutor helpright-arrow

Posted 2 months ago

Question
Evaluate the integral.\newline∫x53xdx \int x 5^{3 x} d x \newlineAnswer:
Get tutor helpright-arrow

Posted 2 months ago

Question
Evaluate the integral.\newline∫−2xe2xdx \int-2 x e^{2 x} d x \newlineAnswer:
Get tutor helpright-arrow

Posted 2 months ago

Question
Evaluate the integral.\newline∫−2xcos⁡(−2x)dx \int-2 x \cos (-2 x) d x \newlineAnswer:
Get tutor helpright-arrow

Posted 2 months ago

Question
Evaluate the integral.\newline∫−xcos⁡(−3x)dx \int-x \cos (-3 x) d x \newlineAnswer:
Get tutor helpright-arrow

Posted 2 months ago

Question
Evaluate the integral.\newline∫6x253xdx \int 6 x^{2} 5^{3 x} d x \newlineAnswer:
Get tutor helpright-arrow

Posted 2 months ago

Question
Evaluate the integral.\newline∫−6x4−4xdx \int-6 x 4^{-4 x} d x \newlineAnswer:
Get tutor helpright-arrow

Posted 2 months ago

Question
Evaluate the integral.\newline∫6x223xdx \int 6 x^{2} 2^{3 x} d x \newlineAnswer:
Get tutor helpright-arrow

Posted 2 months ago

Question
Evaluate the integral.\newline∫−3xsin⁡(−2x)dx \int-3 x \sin (-2 x) d x \newlineAnswer:
Get tutor helpright-arrow

Posted 2 months ago

View all (140)