Evaluate the integrals in Exercises 39–56.
56. ∫sec(x)dx...

Question

Evaluate the integrals in Exercises 39–56.

56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))

Accepted Answer

Welcome back everyone. Evaluate the integral sequent x tangent x dx divided by the square root of e to the power of sequent x. We're going to approach this problem using U substitution. Let's suppose that U is equal to e raised to the power of sequent x. Let's remember that the next step is to differentiate you with respect to x. So DU divided by DX is going to be the derivative of E to the power of sequent x. What we want to do is apply the chain rules. So first of all, we get e to the power of sink and x multiplied by. Second x derivative, right? And now this is equal to e to the power of sequent x multiplied by sequent x tangent x. Now notice that we can rearrange this relationship in terms of sequent x multiplied by tangent x multiplied by dx because this is what we have in the numerator. So what we want to do is simply divide the DU by E to the power of 2 x. And the latter is equal to you, so that's basically DU divided by you. And then we can rewrite our integral as follows. In the numerator we have 2 x tangent x dx, which is d u divided by u. And in the denominator we have square root of e to the power of c and x, which is the square root of u. So now this means we're integrating DU divided by. You square root of you. Now to apply the power rule, we're going to integrate the u divided by u to the power of 1 + 1/22, right? We have u to the power of 1 multiplied by u to the power of 1/2, so the overall exponent is going to be u to the power of 3 halves in the denominator. And finally, we can apply the reciprocal rule to rewrite it as u to the power of -3 halves the u. From here we can apply the power rule which gives us u to the power of -3 halves plus 1 that's u to the power of -1/22 divided by -1/22 plus a constant of integration c. And now we want to simplify, right? So we get -2, 1st of all. Because we're dividing by 1/2 u to the power of -1/2, it can be written in the denominator as square root of u plus c. And now we can rewrite the final answer as -2 divided by square root of in terms of x, the original variable u is e to the power of sequent x. Plus c. So the final answer to this problem. is going to be -2 divided by the square root of e to the power of 2 x + c. Thank you for watching.

Evaluate the integrals in Exercises 39–56.56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))

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Evaluate the integrals in Exercises 39–56.
56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))