Owen Chadwick
06/30/2023 · Senior High School
1. Evaluate the following: Hint: Inverse trigonometry (a) (2 points) \[ \int \frac{1}{x \sqrt{4 x^{2}-2}} d x \]
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Step-by-step Solution
Calculate the integral \( \int \frac{1}{x \sqrt{4 x^{2}-2}} dx \).
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
\(\int \frac{1}{x\sqrt{4x^{2}-2}} dx\)
- step1: Evaluate the power:
\(\int \frac{1}{x\left(4x^{2}-2\right)^{\frac{1}{2}}} dx\)
- step2: Rewrite the expression:
\(\int \frac{1}{x\times 4^{\frac{1}{2}}\left(x^{2}-\frac{1}{2}\right)^{\frac{1}{2}}} dx\)
- step3: Rewrite the expression:
\(\int \frac{1}{2}\times \frac{1}{x\left(x^{2}-\frac{1}{2}\right)^{\frac{1}{2}}\left(x^{2}-\frac{1}{2}\right)^{\frac{1}{2}}} dx\)
- step4: Evaluate the integral:
\(\frac{1}{2}\times \int \frac{1}{x\left(x^{2}-\frac{1}{2}\right)^{\frac{1}{2}}\left(x^{2}-\frac{1}{2}\right)^{\frac{1}{2}}} dx\)
- step5: Evaluate the integral:
\(\frac{1}{2}\times \frac{2}{\sqrt{2}}\times \operatorname{arcsec}\left(\sqrt{2}\times \left|x\right|\right)\)
- step6: Calculate:
\(\frac{1}{\sqrt{2}}\times \operatorname{arcsec}\left(\sqrt{2}\times \left|x\right|\right)\)
- step7: Rewrite the expression:
\(\frac{\operatorname{arcsec}\left(\sqrt{2}\times \left|x\right|\right)}{\sqrt{2}}\)
- step8: Add the constant of integral C:
\(\frac{\operatorname{arcsec}\left(\sqrt{2}\times \left|x\right|\right)}{\sqrt{2}} + C, C \in \mathbb{R}\)
The result of the integral \(\int \frac{1}{x \sqrt{4 x^{2}-2}} dx\) is \(\frac{\operatorname{arcsec}(\sqrt{2}\times |x|)}{\sqrt{2}} + C\), where \(C\) is an arbitrary constant.
Quick Answer
The integral \(\int \frac{1}{x \sqrt{4 x^{2}-2}} dx\) evaluates to \(\frac{\operatorname{arcsec}(\sqrt{2}\times |x|)}{\sqrt{2}} + C\).
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