Determine the following: ∫0^a (d x)/((a^2+x^2)^2) (Put x=a tanθ)

Name: Problem 29, Chapter 30: Engineering Mathematics
Uploaded: 2022-02-17T18:26:52-08:00
Duration: 3 min 53 s
Channel: M Hassan Anwar
Description: Determine the following: ∫0^a (d x)/((a^2+x^2)^2) (Put x=a tanθ)

step 1 Question number 29 left x is equal to a into 10 theta so that a square plus x square whole squar

Determine the following: ∫0^a (d x)/((a^2+x^2)^2) ...

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Key Concepts

Definite Integration

Definite integration involves evaluating an integral with specified upper and lower limits. After performing an appropriate substitution, these limits must be transformed to correspond to the new variable, ensuring that the evaluation of the antiderivative is done over the correct interval.

Trigonometric Identities

Utilizing identities such as tan²(?) + 1 = sec²(?) allows for the simplification of expressions that emerge during trigonometric substitution. These identities are crucial for reducing complex trigonometric expressions into forms that are more readily integrated.

Trigonometric Substitution

This technique involves substituting a trigonometric function for a variable in an integral to simplify expressions, particularly those involving square roots or quadratic forms. In many cases, terms such as a² + x² can be expressed in terms of secant functions using substitutions like x = a·tan(?), leading to more manageable integrals by taking advantage of trigonometric identities.

Change of Variables in Integration

This is a fundamental method in calculus that involves substituting one variable for another to simplify the integral. By rewriting the integral in terms of a new variable and appropriately transforming the differential and limits in definite integrals, the integrand often becomes easier to integrate.

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