Evaluate the integrals in Exercises 91–102.
102. ∫(from ...

Question

Evaluate the integrals in Exercises 91–102.

102. ∫(from -1/3 to 1/√3)(cos(arctan 3x))/(1+9x²) dx

Accepted Answer

Welcome back everyone. Evaluate the integral from 0 to 1/4 of sine of inverse of tangent of 4 x dx divided by 1 + 16 x 2. Let's begin by rewriting the integral. What we're going to do is write it as integral from 0 to 1/4. We're going to leave the numerator as it is, so that we sign off. Inverse of tangent of 4 x followed by d x and in the denominator we're going to apply the properties of exponents and rewrite 1 + 16 x 21 + 4 x squared, right? we can take out that square. And 16 x squared would essentially be 4 x squared. From here we are going to introduce u substitution. Let's suppose that u is equal to inverse of tangent of 4 x. Remember that the next step is to find the derivative of u with respect to x, and here we want to apply the chain rule, so we differentiate inverse of tangent with respect to 4 x, which gives us 1 divided by 1 plus 4 x squared, and according to the chain rule we are multiplying by the derivative of 4 x, which is basically 4 right. From here if we multiply both sides by d x and divide both sides by 4, we will get 1/4. D u is equal to dx divided by 1 + 4 x 2. Then this is exactly what we have within our integral, right? Dx divided by 1 + 4 x 2. So what we have to do now is change the limits of integration. The lower 1 u1 is going to be. Inverse of tangent of 4 multiplied by 0, which is inverse of tangent of 0, and that's 0, and the upper limit of integration u2 is going to be inverse of tangent of 4 multiplied by 1/4. That's inverse of tangent of 1, right, and that's pi divided by 4. So now we can rewrite our integrals integral from 0 to pi divided by 4 of sin. Of u followed by dx divided by 1 + 4 x 2, which is 1/4 d u. So we can just write DU and factor out 1/4 because it's a constant. Well done, and we end up with an elementary integral. We get 1/4, and the integral of sin is negative cosine. So overall we will get 1/4 cosine of u evaluated from 0 to pi divided by 4. And now we have to apply. The fundamental theorem of calculus part 2, which gives us -14, cosine of pi divided by 4 minus. 1/4 cosine of 0. Let's simplify. And what we can do is simply write it as 14, right, negative of negative gives us positive. Cosine of 0 is 1, so that's how we get 1/4 minus 1/4 multiplied by cosine of pi divided by 4 is square root of 2 divided by 2. Now, let's go ahead. And find the common denominator. We have 1/4 minus square root of 2 divided by 8. So the common denominator is 8. We're going to have 2 minus square root of 2 divided by 8 as our final answer. Thank you for watching.

Evaluate the integrals in Exercises 91–102.102. ∫(from -1/3 to 1/√3)(cos(arctan 3x))/(1+9x²) dx

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Evaluate the integrals in Exercises 91–102.
102. ∫(from -1/3 to 1/√3)(cos(arctan 3x))/(1+9x²) dx