Problem 45 - Calculus Early Transcendentals

Question

Evaluate the following integrals. $$\int \frac{2 x^{2}+5 x+5}{(x+1)\left(x^{2}+2 x+2\right)} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral of the following function: $$\int \frac{2x^2+5x+5}{(x+1)(x^2+2x+2)} dx$$ Answer: $$\int \frac{2 x^{2}+5 x+5}{(x+1)\left(x^{2}+2 x+2\right)} d x = \frac{1}{2} \ln|x^2+2x+2| + C$$

Step 1: Perform partial fraction decomposition

The integrand can be rewritten as the sum of simpler fractions: $$\frac{2x^2+5x+5}{(x+1)(x^2+2x+2)} = \frac{A}{x+1} + \frac{Bx+C}{x^2+2x+2}$$ Now, we must find the constants A, B, and C. First, multiply both sides by the common denominator \((x+1)(x^2+2x+2)\): $$2x^2 + 5x + 5 = A(x^2+2x+2) + (Bx+C)(x+1)$$ We can substitute the values that make the denominators zero to find the constants: For A, let x=-1: $$2(-1)^2 + 5(-1) + 5 = A((-1)^2+2(-1)+2)$$ $$5 - 5 = 2A$$ $$A = 0$$ For B and C, we can expand the equation and collect similar terms: $$2x^2 + 5x + 5 = (Bx^2 + Bx + Cx + C)$$ Comparing the coefficients, we have: $$B = 2$$ $$B + C = 5$$ Solving for C: $$C = 5 - B$$ $$C = 5 - 2$$ $$C = 3$$ Now, we have found the constants A, B, and C, and our integrand becomes: $$\frac{2x^2+5x+5}{(x+1)(x^2+2x+2)} = \frac{2x+3}{x^2+2x+2}$$

Step 2: Integrate each of the simpler fractions

Now, we can integrate each term separately: $$\int \frac{2x^2+5x+5}{(x+1)(x^2+2x+2)} dx = \int \frac{2x+3}{x^2+2x+2} dx$$ To integrate the second fraction, we substitute: $$u = x^2 + 2x + 2$$ $$\frac{du}{dx} = 4x + 2$$ Therefore, $$dx = \frac{du}{4x+2}$$ and the integral becomes: $$\int \frac{2x+3}{u} \times \frac{du}{4x+2}$$ Now, we can rewrite our integral using the substitution: $$\int \frac{2x+3}{u} \times \frac{1}{2}du = \frac{1}{2}\int \frac{1}{u}du$$

Step 3: Combine the results from step 2 to obtain the final solution

Now we can integrate the new function: $$\frac{1}{2}\int \frac{1}{u}du = \frac{1}{2} \ln|u| + C$$ Finally, substitute back for u: $$\frac{1}{2} \ln|x^2+2x+2| + C$$ Thus, the integral of the given function is: $$\int \frac{2 x^{2}+5 x+5}{(x+1)\left(x^{2}+2 x+2\right)} d x = \frac{1}{2} \ln|x^2+2x+2| + C$$

Key Concepts

Calculus

Calculus is a branch of mathematics that studies continuous change, and it is split into two main fields: differential calculus and integral calculus. Differential calculus concerns itself with the concept of a derivative, which represents the rate at which a function is changing at any given point. Integral calculus, on the other hand, deals with the integration of functions, which is essentially the process of finding the quantity where the rate of change is known.

In the context of our integral exercise, we use integral calculus to find the area under the curve defined by the function \( \frac{2x^2+5x+5}{(x+1)(x^2+2x+2)} \). This process involves finding a function known as the antiderivative, which, when differentiated, gives us our original function.

Integration

Integration, a core concept of calculus, refers to the process of calculating the integral of a function. This process can be thought of as finding the total accumulation of quantity, such as area under a curve or the total distance traveled given a speed function. There are several techniques for performing integration, and one particularly useful method is 'integration by substitution', often applied when a function is too complex to integrate in its original form.

In the given exercise, integration by substitution proves helpful for simplifying the second fraction \( \frac{Bx+C}{x^2+2x+2} \) after decomposition. By substituting \( u = x^2+2x+2 \) and \( du = (4x+2)dx \), the integral becomes much more manageable.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying mathematical expressions and equations. It is a fundamental skill in calculus, often used prior to the application of various calculus techniques. The purpose of this manipulation is to transform an equation or expression into a form that is easier to work with or solve.

In the provided exercise, algebraic manipulation is crucial in the first step, where partial fraction decomposition requires us to express the integrand as a sum of simpler fractions. We then determine the coefficients for these fractions through further algebraic manipulation by equating coefficients and solving the resulting system of equations.

Integral Calculus

Integral calculus is one of the two main divisions of calculus, focusing on the concept of an integral. It enables the computation of properties such as areas, volumes, and accumulation of quantities. The fundamental theorem of calculus links integral calculus with differential calculus, showing that integration is the reverse process of differentiation.

In our integral problem, once we have completed the partial fraction decomposition, we apply techniques of integral calculus to find the antiderivative of the simpler fractions. By finding the antiderivative of \( \frac{2x+3}{x^2+2x+2} \) after an appropriate substitution, we are able to solve the integral and thus calculate the area under the original function's curve.