Integrating Complex Functions: A Step-by-Step Guide Using Symmetry and Substitution

In this article, we provide a detailed guide on how to integrate a complex function with the help of symmetry and substitution. We will walk through a specific example to demonstrate the process.

Understanding the Problem

Consider the integral:

I int_{3}^{7} (x-37-x)^{1/4} dx

Our goal is to find I by simplifying the expression inside the integral using symmetry and substitution techniques.

Step 1: Use a Substitution

First, notice the symmetry of the function (x-37-x)^{1/4} around the midpoint of the interval [3,7]. This midpoint is at x 5.

Let x 5u 5. Then, the new limits are:

When x 3, u (3-5)/5 -2/5 When x 7, u (7-5)/5 2/5

Add these to the integral, we get:

I int_{-2}^{2} (5u-37-5u)^{1/4} du

Step 2: Simplify the Integrand

Evaluate the expression inside the integral:

(5u-37-5u) (2u^2 - u)

Further simplify:

(2u^2 - u) 4 - u^2

Therefore, the integral simplifies to:

I int_{-2}^{2} (4 - u^2)^{1/4} du

Step 3: Use Symmetry

Notice that the integrand (4 - u^2)^{1/4} is an even function. We can reduce the integral to half of the interval:

I 2 int_{0}^{2} (4 - u^2)^{1/4} du

Step 4: Evaluate the Integral

Let u 2sin(theta), then du 2cos(theta)dtheta. The limits change as follows:

When u 0, theta 0 When u 2, theta pi/2

We get:

J int_{0}^{pi/2} (4 - 2sin^2(theta))^{1/4} cdot 2cos(theta) dtheta

Further simplify:

J int_{0}^{pi/2} (2(1 - sin^2(theta)))^{1/4} cdot 2cos(theta) dtheta int_{0}^{pi/2} (2cos^2(theta))^{1/4} cdot 2cos(theta) dtheta

Since cos^2(theta) cos^{2} (theta) we get:

J 2^{3/4} int_{0}^{pi/2} cos^{3/2}(theta) dtheta

Step 5: Evaluate int_{0}^{pi/2} cos^{3/2}(theta) dtheta

Using the Beta function or Gamma function:

int_{0}^{pi/2} cos^n(theta) dtheta frac{Gamma(frac{n 1}{2})}{Gamma(frac{n}{2} 1)}

For n 3/2 we have:

int_{0}^{pi/2} cos^{3/2}(theta) dtheta frac{Gamma(1.25)}{Gamma(1.75)}

Using the values of the Gamma function:

Gamma(1.25) frac{1}{4}sqrt{pi}

Gamma(1.75) frac{3}{4}sqrt{pi}

Therefore:

int_{0}^{pi/2} cos^{3/2}(theta) dtheta frac{frac{1}{4}sqrt{pi}}{frac{3}{4}sqrt{pi}} frac{1}{3}

Step 6: Final Result

Putting it all together:

J 2^{3/4} cdot frac{1}{3} frac{2^{3/4}}{3}

Recall that I 2J 2 cdot frac{2^{3/4}}{3} frac{2^{7/4}}{3}

Therefore, the final result for the integral is:

I frac{2^{7/4}}{3}

Conclusion

In this article, we have demonstrated a systematic approach to integrating complex functions using symmetry and substitution. By applying these techniques, we were able to simplify the integral and find its value.