Symmetry in integrals Use symmetry to evaluate the following i...

Question

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫₋π/₄^π/⁴ sec² x dx

Accepted Answer

Welcome back, everyone. Evaluate the integral from negative pi divided by 6 to pi divided by 6 of 5c2XDX using symmetry. So first of all, let's consider our function. We can rewrite our integral by factoring out the constant, which is 5 multiplied by the integral from negative pi divided by 6 to pi divided by 6 of sequent squared XD X, and the function that we're considering is squad of X. So let's go ahead and evaluate squad of negative X to classify this function as even or odd. And we know that sant is basically 1 divided by cosine, so we're evaluating 1 divided by cosine of negative X, and we're squaring the result. Let's recall that cosine is an even function. So essentially, cosine of negative X is, cosine X. And now we have shown that we end up with one divided by a cosine of X. All fraction squared, which is basically c 2d X, right, and because c 2d of negative X is equal to c2X, this is an even function, meaning it is symmetric relative to the y axis and therefore we can say that our integral can be rewritten as 5 multiplied by 2. Integral from 0 to pi divided by 6 of squared XDX based on symmetry. So basically we're taking our integral and replacing. The interval from negative pi divided by 6 up to divided by 6 with 0 up to pi divided by 6 and multiplying it by 2, right? This is the property for even functions. And now we can evaluate our integral. Let's label it as I. That'll be 10 integral from 0 to pi divided by 6 of c squad. Of XC X and we have to recall that the integral of c squared is tangent. So we are evaluating 10 multiplied by Tangent of X from 0 to pi divided by 6. Applying the limits we get 10 inferences tangent of pi divided by 6 minus tangent of 0. Now a tangent of 0 is 0, so we basically get 10 multiplied by tangent of pi divided by 6, which is square root of 3 divided by 3. So the final answer for this problem would be 10 square root of 3 divided by 3. Thank you for watching.

Symmetry in integrals Use symmetry to evaluate the following integrals.
∫₋π/₄^π/⁴ sec² x dx

Key Concepts

Symmetry in Integrals

Secant Function

Definite Integrals

Watch next