7-56. Trigonometric substitutions Evaluate the following integ...

Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

46. ∫ 1/√(1 - 2x²) dx

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to evaluate the integral using trigonometric substitution, and we're given the integral of 1 divided by the square root of the quantity of 1 minus 3 X squared in quantity DX. So we need to evaluate our integral here using trigonometric substitution. So we were given the integral of 1 divided by the square root of the quantity of 1 minus 3 x 2 and quantity DX. And the trick substitution that we want to use is we want to set X equal to 1 divided by the square root of 3, multiplied by sine of data. And using this definition for X, we'll see that DX is going to be equal to 1 divided by the square root of 3, multiplied by cosine of theta D theta. So this means that our interval is going to be equal to. The integral of 1 divided by the square root of 3, multiplied by cosine of data. This can be divided by the square root of the quantity of 1 minus 3 multiplied by the quantity of 1 divided by the square root of 3, multiplied by the sign of data in quantity squared in quantity the theta. So now we just need to simplify this expression. So we'll factor the constant of 1 divided by the square root 3 out in front of the integral. So this is going to be equal to 1 divided by the square root of 3, multiplied by the integral, and in the numerator we'll just have cosine of data, and this is going to be divided by the square root of the quantity of 1 minus, and here we have 3 divided by the square root of 3 squad, which is just 1. And then that gets multiplied by sine squared data. So we'll have 1 minus sine quad data underneath the radical, and then all that is still multiplied by the theta. Now, we can use our trig identities to rewrite our denominators. So recall that 1 minus sine squared data is equal to cosine squared data. That means that this is going to be equal to 1 divided by square root of 3, multiplied by the integral of cosine of data. Divided by, and here we have the square root of cosine square of theta, which is just cosine of theta, and that gets multiplied by the theta still. So here we have cosine theta divided by cosine theta, which is 1. So this is just going to be equal to 1 divided by the square root of 3, multiplied by the integral of the data. And so performing this integration, we see that this is going to be equal to 1 divided by the square root of 3, multiplied by theta plus C since we need to include our integration constant. So now what we need to do is rewrite data in terms of X. So using our definition from X earlier, we see that data. It's going to be equal to the inverse sign. Of the quantity of the square root of 3 multiplied by X in quantity. So making this substitution. Into our results here we see that our original integral, which is the integral of 1 divided by. The square root of the quantity of 1 minus 3 x 2 in quantity d X is going to be equal to 1 divided by the square root of 3 multiplied by the inverse sign of quantity of the square root of 3 multiplied by X in quantity plus D. 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.

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
46. ∫ 1/√(1 - 2x²) dx

Key Concepts

Trigonometric Substitution

Pythagorean Identity

Integral Evaluation

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