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Show the integral in which the substitution x = 7/2tan(t) transforms
I =∫dx/√49x² + 4

User GaryO
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Final answer:

Using the substitution x = 7/2tan(t), we can transform the integral I = ∫dx/√(49x² + 4) into a form involving trigonometric identities, which assists in simplifying and calculating the integral.

Step-by-step explanation:

The substitution x = \(\frac{7}{2}\)tan(t) is used to simplify the integral I = \int\frac{dx}{\sqrt{49x^2 + 4}}. To perform this substitution, we will need to also find the differential dx in terms of dt. Using the trigonometric identity for tan(t), this substitution will lead to a more manageable form of the integral. The steps are as follows:

  • Determine dx by differentiating x with respect to t.
  • Substitute x and dx into the original integral.
  • Use the Pythagorean identity to simplify the resulting trigonometric expression.
  • Integrate with respect to t.

By doing so, we leverage trigonometric identities and substitutions to evaluate integrals that are otherwise difficult to compute directly.

User Abadis
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