96. Challenge
Show that with the change of variables u =...

Question

96. Challenge

Show that with the change of variables u = √tan x, the integral

∫ √tan x dx

can be converted to an integral amenable to partial fractions. Evaluate

∫[0 to π/4] √tan x dx.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to express the interval of the cube root of tangent of XDX in a form suitable for partial fraction decomposition by using the substitution U is equal to the cube root of tangent of X. So to solve this problem, the first step is to use our substitution. U is equal to the cube root of tang of x to find DX in terms of U and DU. So the first step is to cube both sides of the equation for our substitution. In doing so, we'll get that u cubed is equal to tangent of X. And what we're going to do is differentiate both sides of this equation. In doing so, we'll get that 3 U squared, DU. It's going to be equal to. Tin squared of X. DX. However, we can rewrite squad of X in terms of the tangent of X using our trig identities. So that means that 3 U 2 DU is going to be equal to the quantity of 1 plus tangent squared of X in quantity DX. However, tangent of X is equal to U cubed. That means that 3 U 2 DU is going to be equal to the quantity of 1 plus 6 in quantity DX. And the sulfur DX we'll just divide both sides of this equation by the quantity of 1 plus U 6 in quantity. In doing so, we'll get that DX. It's going to be equal to 3 U 2 divided by the quantity of 1 plus U 6 in quantity DU. So coming back to our original expression, which is going to be the interval of the cube root of tangent of X D X, we see that this is going to be equal to the integral of and here we'll substitute in for the cube root of tangent of X, our value of U, so we'll have U multiplied by DX, so we'll substitute in the expression that we just found for DX, which is going to be the quantity of 3 U squared. Divided by the quantity of 1 plus U 6 and quantity D U. This is just multiplying the two terms in our anagram together. We see that this is going to be equal to. The integral of 3 u cubed divided by the quantity of 1 plus U 6 in quantity D U. Now, if we look at our integram, we see that it's a rational function where the degree of the numerator is less than the degree of the denominator. So this makes it suitable for partial fraction decomposition. So that means that this here is going to be the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

96. Challenge
Show that with the change of variables u = √tan x, the integral
∫ √tan x dx
can be converted to an integral amenable to partial fractions. Evaluate
∫[0 to π/4] √tan x dx.

Key Concepts

Change of Variables (Substitution)

Partial Fraction Decomposition

Definite Integration with Trigonometric Limits

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