Evaluate the given integral by making a trigonometric substitution (even if you spot another way

Evaluate the given integral by making a trigonometric...

Key Concepts

Substitution Method in Integration

The substitution method is a fundamental technique in integration that involves changing the variable of integration to simplify the integral. This process includes substituting both the variable and its differential, which can transform a difficult integral into one that is more straightforward by aligning it with standard integral formulas or identities.

Integration of Radical Expressions

Integrating expressions that involve radicals often requires specialized techniques to handle the square roots of polynomial functions. These integrals can be complex due to the non-linear nature of the square root function, and transforming the integrand through methods like substitution is a common strategy to facilitate their evaluation.

Trigonometric Substitution

Trigonometric substitution is a method used to simplify integrals involving radical expressions by substituting variables with trigonometric functions. By doing so, expressions under the square root are transformed using trigonometric identities into forms that are easier to integrate. This technique is particularly useful when the integrand contains expressions like ?(a² + x²), ?(a² - x²), or ?(x² - a²).

Transcript

00:02 In this problem, we have to find integral of 4 times square root of 1 plus 4x squared d of x.

00:21 Now, we are going to evaluate this integral using trigonometric substitutions.

00:30 So let us consider x is equal to 1 over 2 tangent.

00:36 Of theta because for this substitution we get rid from the square root in the integrant now taking differential on both sides we get d of x is equal to 1 over 2 times differential of tangent theta will be second square theta d of theta now using these substitutions in the above integral we have taking 4 outside the integral sign we have 1 plus 4 times x square will be 1 over 4 tangent square theta whole square root times d x is 1 over 2 second square theta d of theta now this will be equal to 4 times integer of square root of here 4 4 get cancelled and we have 1 plus tangent square theta which will be equal to second square theta times here we have 1 over 2 times second square theta d of theta now taking 1 over 2 outside day the integral sign we have 4 times 1 over 2 will be 2 times integral of here square and root cancelled and we have second theta times second square theta which will be equal to second cube theta d theta now this can be written as two times integral of second theta times second square theta d theta.

02:58 Here we are going to use integration by parts taking second theta as first function and second square theta as the second function.

03:11 Now this will be equal to two times.

03:16 Here we are going to use integration by parts.

03:23 So first function, second theta, integral of the second function, that is integral of second square theta will be tangent of theta minus integral into derivative of the first function, that is derivative of second theta will be second theta, 10 theta, and integral of the second function will be tangent of theta d of theta.

04:03 Now this will be equal to 2 second theta tangent of theta minus two times integral of second theta tangent square theta d of theta...

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