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

Verified step by step guidance

1. Step 1: Recognize that the integrand (x+2)/(x+4) can be simplified using substitution. Let u = x + 4, which implies du = dx.
2. Step 2: Rewrite the numerator in terms of u. Since u = x + 4, we can express x + 2 as (u - 2). Substitute this into the integrand, yielding ∫(u - 2)/u du.
3. Step 3: Split the integrand into two simpler fractions: ∫(u/u - 2/u) du = ∫1 du - ∫(2/u) du.
4. Step 4: Evaluate each term separately. The integral of 1 with respect to u is u, and the integral of 2/u with respect to u is 2ln|u|.
5. Step 5: Substitute back u = x + 4 into the result to express the solution in terms of x. The final expression will be x + 4 - 2ln|x + 4| + C, where C is the constant of integration.

Key Concepts

Integration Techniques

Integration techniques are methods used to find the integral of a function. In this case, recognizing that the integrand can be simplified using substitution or properties of logarithms can help evaluate the integral without long division. Understanding these techniques is crucial for efficiently solving integrals.

Substitution Method

The substitution method involves changing the variable of integration to simplify the integral. For the given integral, one might let u = x + 4, which transforms the integrand into a more manageable form. This technique is particularly useful when the integrand can be expressed in terms of a single variable.

Properties of Logarithms

Properties of logarithms can be applied when integrating rational functions. Specifically, the integral of a function of the form (a + bx)/(c + dx) can often be expressed in terms of logarithmic functions. Recognizing these properties allows for a quicker evaluation of integrals without resorting to polynomial long division.