7–64. Integration review Evaluate the following integrals.
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Question

7–64. Integration review Evaluate the following integrals.

60. ∫ from 0 to π/4 of 3√(1 + sin 2x) dx

Accepted Answer

Welcome back, everyone. Evaluate the integral. Integral from 0 to pi divided by 2 of 2 square root of 1 plus sine 2 X D X A 0, B1, C2, and D 4. So for this problem, let's rewrite our integral. We have integral from 0 to pi divided by 2 of 2, so we can factor out the constant square root of. 1 plus sign of 2 XDX. What we're going to do is simply rewrite sine of 2 X using the double angle formulas. We got 2 multiplied by the integral from 0 to pi divided by 2, and under the radical. We can perform several changes. First of all, let's rewrite one using the Pythagorean identity, and that would be sin squared X plus cosine squared X, right? This is equal to 1. Now, for sine of 2 X, applying the double angle formula, we get 2 sine x cosine x. So this is the expanded form under the radical, and what we can notice is that it is a perfect square. We can write it in a form of a sum. Which is squared, right? So we get 2 multiplied by the integral from 0 to pi divided by 2 of square root of. And now let's notice that this is in the form of A2 + 2 AB plus B2, meaning under the radical we can write A + B, which is sin x plus cosine x. And this sum is squared. And then we are adding DX. So now what we want to do is simply take square root of that, and now let's be careful and remember that square root of X squad is the absolute value of X, right? It can be positive or negative X. So what we want to do is simply analyze the limits of integration. And notice that we're in the first squadron because X belongs to the interval from 0 to pi divided by 2, and in the first squadron, both sine and cosine, they are non-negative, right? So essentially this means that their sum is going to be non-negative as well. So essentially we're going to take the positive value, meaning our integral becomes 2 integral from 0. To pi divided by 2 of sine x plus cosine x. D X And now we are ready to integrate, so that'll be 2. The integral of sine is negative cosine, so we get negative cosine X, and the integral of cosine is sign, so we're adding sin X. And now our goal is to evaluate from 0 to pi divided by 2. Which is equal to Let's value the result at pi divided by 2, so that'd be negative cosine. Of pi divided by 2 plus sign of pi divided by 2. And then we are subtracting. 2 Multiplied by negative cosine at 0, plus sign at 0. An outlets simplify. So, cosine of pi divided by 2 is going to be equal to 0. Sign of pi divided by 2 is equal to 1, cosine of 0 is equal to 1, and sine of 0 is 0, meaning we get 2 multiplied by 1. -2 multiplied by -1. Which is basically 2 + 2 and that's 4, meaning the correct answer to this problem corresponds to the answer choice D. Thank you for watching.

7–64. Integration review Evaluate the following integrals.
60. ∫ from 0 to π/4 of 3√(1 + sin 2x) dx

Key Concepts

Definite Integrals

Trigonometric Identities

Substitution Method

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