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I=int(x)/(sqrt(x^(2)-7))dx

I=∫xx2−7dx I=\int \frac{x}{\sqrt{x^{2}-7}} d x

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Full solution

Q. I=∫xx2−7dx I=\int \frac{x}{\sqrt{x^{2}-7}} d x
  1. Substitution and Simplification: Now, we substitute x=7sec⁡(θ)x = \sqrt{7}\sec(\theta) and dx=7sec⁡(θ)tan⁡(θ)dθdx = \sqrt{7}\sec(\theta)\tan(\theta)d\theta into the integral:\newlineI=∫xx2−7dxI = \int\frac{x}{\sqrt{x^2 - 7}}dx\newlineI=∫7sec⁡(θ)72sec⁡2(θ)−7(7sec⁡(θ)tan⁡(θ)dθ)I = \int\frac{\sqrt{7}\sec(\theta)}{\sqrt{7^2\sec^2(\theta) - 7}}(\sqrt{7}\sec(\theta)\tan(\theta)d\theta)
  2. Further Simplification: Simplify the expression inside the square root and the integral:\newlineI=∫7sec⁡(θ)72(sec⁡2(θ)−1)(7sec⁡(θ)tan⁡(θ) dθ)I = \int \frac{\sqrt{7}\sec(\theta)}{\sqrt{7^2(\sec^2(\theta) - 1)}}(\sqrt{7}\sec(\theta)\tan(\theta)\,d\theta)\newlineI=∫7sec⁡(θ)72tan⁡2(θ)(7sec⁡(θ)tan⁡(θ) dθ)I = \int \frac{\sqrt{7}\sec(\theta)}{\sqrt{7^2\tan^2(\theta)}}(\sqrt{7}\sec(\theta)\tan(\theta)\,d\theta)
  3. Integration: Since tan⁡2(θ)\tan^2(\theta) is the result of sec⁡2(θ)−1\sec^2(\theta) - 1, we can further simplify the integral:\newlineI=∫7sec⁡(θ)7tan⁡(θ)(7sec⁡(θ)tan⁡(θ)dθ)I = \int\frac{\sqrt{7}\sec(\theta)}{7\tan(\theta)}(\sqrt{7}\sec(\theta)\tan(\theta)d\theta)\newlineI=∫sec⁡2(θ)tan⁡(θ)dθI = \int \sec^2(\theta)\tan(\theta)d\theta
  4. Integration of sec⁡2(θ)tan⁡(θ)\sec^2(\theta)\tan(\theta): Now, we can integrate sec⁡2(θ)tan⁡(θ)\sec^2(\theta)\tan(\theta) with respect to θ\theta: \newlineI=∫(sec⁡2(θ)tan⁡(θ)dθ)I = \int(\sec^2(\theta)\tan(\theta)d\theta)\newlineI=sec⁡(θ)2+CI = \frac{\sec(\theta)}{2} + C, where CC is the constant of integration.
  5. Reverting to Original Variable: We need to revert back to the original variable xx. From our substitution, we have x=7sec⁡(θ)x = \sqrt{7}\sec(\theta), so sec⁡(θ)=x7\sec(\theta) = \frac{x}{\sqrt{7}}. Substituting this back into our integral gives us:\newlineI=(x7)2+CI = \frac{\left(\frac{x}{\sqrt{7}}\right)}{2} + C\newlineI=x27+CI = \frac{x}{2\sqrt{7}} + C
  6. Final Indefinite Integral: Finally, we have the indefinite integral of the function f(x)=xx2−7f(x) = \frac{x}{\sqrt{x^2 - 7}} in terms of xx:I=x27+CI = \frac{x}{2\sqrt{7}} + C

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