70. Different methods Let I=∫(x+2)/(x+4)dx.
b. Evaluate ...

Question

70. Different methods Let I=∫(x+2)/(x+4)dx.

b. Evaluate I without performing long division on the integrand.

Accepted Answer

Welcome back, everyone. Find K equals integral of X + 3 divided by X + 6DX without using long division. For this problem, we have a ratio of two linear functions within our integral. So what we're going to do is simply use U substitution for the bottom function. Let's suppose that U equals x + 6. We want to identify the derivative the u divided by DX, which is 1. Meaning DU equals the X, right? And then we want to rewrite our integral K. So integral K becomes now X + 3. Simply becomes. Now, what we have to do is solve for X, right? X is equal to U minus 6, so we can now express X in the numerator. We get u minus 6 + 3. Considering the denominator, we have X plus 6, which is EU itself, right? And the X becomes DU. Now we can combine the like terms, U minus 6 + 3 is U minus 3. We are dividing that by U and multiplying by the u. Now we can divide, so you divided by you is 1. And we're subtracting we divided by you, right? And then we can split it into two separate integrals. So the first one is going to be integral of the U. And for the second one, we can factor out 3 and multiply it by the integral of 1 divided by U, or simply the U divided by you. Let's valuate each integral. The integral of the U is simply U. We're subtracting 3 which is multiplied by the integral of the u divided by U. That's a basic integral Ln, the absolute value of U, and we want to add a constant of integration C because this is an indefinite integral. And now substituting back since U equals x + 6, we get X + 6 minus 3 lm off. X + 6. I of the absolute value of X plus 6, I'm sorry, plus C, right? And then what we notice is that our constant of integration C and 6, they can be combined into a constant of integration, right? Because 6 is a constant. So to avoid confusion, we can just say that previously we got a constant of integration C1. And now it becomes an overall constant C, so the final answer would be X minus 3 lm, the absolute value of X plus 6 + C. This is our final answer and thank you for watching.

70. Different methods Let I=∫(x+2)/(x+4)dx.
b. Evaluate I without performing long division on the integrand.

Key Concepts

Integration Techniques

Substitution Method

Properties of Logarithms

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