7–84. Evaluate the following integrals.
82. ∫ 1/(1 + tan...

Question

7–84. Evaluate the following integrals.

82. ∫ 1/(1 + tanx) dx

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says to evaluate the indefinite integral, and we're given the integral of 1 divided by the quantity of 1 plus cotangent of X in quantity DX. So, our first step into evaluating this integral is to rewrite cotangent of X in terms of sign and cosine. So we'll have the integral of 1 divided by the quantity of 1 plus, and here we call that cotangent of X is equal to cosine of X. Divided by sine of X. In quantity, DX. And so, we want to remove the fraction of the denominator, so we'll multiply both the numerator and denominator by sign of X. And in doing so, our integral is going to be equal to the integral of sin of X. Divided by the quantity of sin of x. Plus cosine of X. In quantity, DX. Now, this is still not in a form that we can integrate directly. So, what we want to do now is multiply this by 1 in the form of sin of X minus cosine of X, and that quantity is going to be divided by. Sign of X Minus cosine of x. So doing this, Our integral is going to be equal to the integral of sin squared of X minus cosine of X. Multiplied by sin of X. In quantity, divided by the quantity of sine squared of X minus cosine squared of X in quantity DX. So, now we can actually use some trigger identities to rewrite our integrand here. So, recall that sin squared of X minus cosine squad of X is equal to minus cosine of 2 X. And also, we have sin squared of X minus cosine of X multiplied by sin of X. Is equal to 1/2 multiplied by the quantity of 1 minus cosine of 2 X. In quantity Minus 1/2 multiplied by sin of 2 X. So, using these identities, we can rewrite our interval as being equal to the integral. Of the quantity of 1/2 multiplied by the quantity of 1 minus cosine of 2 X. In quantity, minus 12 multiplied by the sine of 2 X. In quantity, divided by the quantity of minus cosine of 2 X in quantity DX. So now we want to do is to split our integrand up into 3 separate fractions. And so we will now have 3 different integrals. So doing so, this is going to be equal to. For the first term, we will have minus 1/2 multiplied by the integral of 1 divided by cosine of 2 X, which is sequent of 2 X. DX Then we'll have plus. One half, multiplied by the integral of cosine of 2 X divided by cosine of 2 X, which is 1. So we'll just have the interval of DX. Then we'll have plus 1/2 multiplied by the integral of, and here we have sin of 2 x divided by cosine of 2 X which will give us tangent of 2 X. Multiplied by DX. So now we just need to evaluate each one of these integrals individually. So this is going to be equal to minus 1/2 multiplied by the quantity, and here when we integrate sequent of 2 X, recall that we're going to get 1 divided by 2 multiplied by the natural log of the absolute value of sequent of 2 X. Plus the tangent of 2 X. In quantity Then we'll have plus 1/2 multiplied by, and here we're integrating 1 by DX which will just give us X. And then, finally, for our last term, we'll have plus 1/2 multiplied by the quantity, and here we're integrating tangent of 2 X, and so recall that this is going to be equal to 1/2 multiplied by the natural log of the absolute value of sequent of 2 X. In quantity And then finally, each one of these integrals will have an integration constant, which we can combine into a single integration constant of plus C at the end. So now we just need to simplify this expression by combining our two natural log terms. In doing so, this is going to be equal to 1/2 multiplied by X. Plus 1 divided by 4 multiplied by the natural log of the absolute value of the quantity of sequent of 2 X. Divided by the quantity of sequant of 2 X. Plus tangent of 2 X. In quantity Plus C. So this here is the answer that we were looking for for this problem. So, I hope that this explanation has been useful, and I'll see you in the next video.

7–84. Evaluate the following integrals.
82. ∫ 1/(1 + tanx) dx

Key Concepts

Integration

Trigonometric Functions

Substitution Method

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