Use any method to evaluate the integrals in Exercises 55–66.

Question

∫ x / (x + √(x² + 2)) dx

Accepted Answer

Hello. In this video, we are going to be evaluating the following integral. The integral given to us is the integral of x divided by the quantity x plus the square root of x2 + 5. Now here, what we can go ahead and do is we can simplify the integrand. One way that we can simplify the integrand is by multiplying by the integrands conjugate. So if we're taking a look at just the integrat itself, we have x divided by x plus the square root of x2 + 5. We can eliminate the square root in the denominator by multiplying by the denominator's conjugate, which is going to be x minus the square root of x2 + 5. Divided by x minus the square root of x2 + 5. Now if we were to multiply these quantities together, we will be left. With -1/5 multiplied by x2 minus x multiplied by the square root of x2 + 5, this is going to be the simplified form of the integrand. So putting this back into the integral is going to leave us with the constant -1/5 multiplied by the integral of x 2. Minus x multiplied by the square root of x2 + 5. And so here what we can go ahead and do is we can go ahead and split this up into a difference of integrals. So we will have 1/5 multiplied by the integral of X2 minus the integral of X multiplied by the square root of X2 + 5. Let's go ahead and label the first integrals integral 1, and the second is integral 2, and we will evaluate them independently. Now for the first integral, the first integral is fairly easy. By applying the standard integral definition, this will integrate to be 1/3 x cubed, and that's going to be the final integran or the final antiderivative of the first integral. Now for the second integral, we have the integral of x multiplied by the square root of X2 + 5. If we allow u to equal to X2 + 5, then DU is equal to 2 XDX, and this is equivalent to 1/2 DU equal to XDX. This will allow us to simplify the 2nd integral to be 1/2 multiplied by the integral of u to the 12 power d u. And the antiderivative is going to leave us with 1/2 multiplied. By 2/3 of u to the 3/2 power. This will further simplify to be 1/3 u to the 3 halves, but then resubstituting back our U substitution will give us X2 + 5 raise to the 3/2 power. So this is going to be the fully evaluated form of the 2nd integral. Now if we were to put this back into the original difference, we will have -15. Multiplied by the quantity. 1/3 x cubed minus 1/3 multiplied by X2 + 53 halves plus C. Now here what we can go ahead and do is we can go ahead and just distribute the -1/5 to each term and so that will leave us with 1 divided by 15 x cubed. Plus 1 divided by 15, multiplied by the quantity X2 + 5, raised to the 3 halves power plus C. This is going to be the final form of the antiderivative, or this is the solution to the given integral, and with that being said, the overall solution to this problem is going to bed. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Use any method to evaluate the integrals in Exercises 55–66.∫ x / (x + √(x² + 2)) dx

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Use any method to evaluate the integrals in Exercises 55–66.
∫ x / (x + √(x² + 2)) dx