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Magazine Chapter 17: Problem 13
Find \(\int \frac{2 x+7}{x^{2}+7 x} \mathrm{~d} x\).
Short Answer
Expert verified
Question: Evaluate the integral \(\int \frac{2x+7}{x^2+7x} dx\).
Answer: \(\int \frac{2x+7}{x^2+7x} dx = \ln|x^2+7x| + C\), where \(C\) is an integration constant.
Step by step solution
01
Identify the substitution
Let's look at our integral,
$$\int \frac{2x+7}{x^2+7x} dx.$$
Notice that if we differentiate the denominator -\(x^2 + 7x\)-, we obtain almost exactly the numerator. Specifically,
$$\frac{d}{dx}(x^2+7x) = 2x + 7.$$
Thus, we will make the substitution \(u=x^2+7x\).
02
Compute du in terms of dx
Now, let's differentiate \(u\) with respect to \(x\) and isolate \(dx\).
$$\frac{du}{dx} = 2x + 7.$$
So,
$$du = (2x + 7)dx.$$
03
Rewrite the integral in terms of u
Now that we have the substitution and its differential, we can rewrite the integral in terms of \(u\):
$$\int \frac{2x+7}{x^2+7x} dx = \int \frac{1}{u} du.$$
04
Evaluate the integral in terms of u
This new integral is straightforward to evaluate:
$$\int \frac{1}{u} du = \ln|u|+C,$$
where \(C\) is an integration constant.
05
Re-substitute in terms of x
Finally, we substitute the original expression for \(u\) back into the integral:
$$\ln|u|+C = \ln|x^2+7x|+C.$$
06
Simplify and write the final answer
Writing the answer more neatly, we get:
$$\int \frac{2x+7}{x^2+7x} dx = \ln|x^2+7x| + C.$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration by Substitution
Integration by substitution is a fundamental technique in calculus used when evaluating integrals. This method is especially useful when an integral contains a composite function.
It's akin to the chain rule for differentiation but applied in reverse. Here鈥檚 how it works:
It's akin to the chain rule for differentiation but applied in reverse. Here鈥檚 how it works:
- First, identify a part within the integrand (the function inside the integral) that can be substituted with a new variable, say, \( u \). This is typically the inner function of a composition.
- Once you have chosen the substitution \( u \), express \( du \) in terms of \( dx \) to transform the entire integral from the variable \( x \) to the variable \( u \) to make integration simpler.
- The goal is to simplify the integrand so it becomes easier to solve using basic integration rules.
- Eventually, after solving the integral with respect to \( u \), revert to the original variable \( x \) by back-substituting the expression for \( u \).
Definite Integrals
Definite integrals allow us to determine the area under a curve between two specific limits. Unlike indefinite integrals which result in a general form or family of functions, definite integrals give us a numerical value.
This is primarily because definite integrals don't just include an integral sign and a differential, but also limits of integration.
This is primarily because definite integrals don't just include an integral sign and a differential, but also limits of integration.
- When working with a definite integral, always ensure that the limits are appropriately transformed if a substitution method is used.
- For example, if \( x \) ranges from \( a \) to \( b \), and we substitute in \( u \), the limits must be transformed accordingly based on the substitution.
- This accurately reflects the same 'slices' of area under the curve, merely expressed in terms of a different variable.
Integration Techniques
There are various techniques for solving integrals, each with its own strengths and ideal applications. Recognizing which method to apply is a key part of mastering calculus.
Several methods include:
Several methods include:
- Integration by Parts: Useful for products of functions. This technique arises from the product rule in differentiation.
- Trigonometric Integrals: Special rules for integrals involving trigonometric functions and identities.
- Partial Fractions: Decomposing rational expressions into simpler fractions before integrating.
- Numerical Integration: Methods like the Trapezoidal rule or Simpson's rule for approximating the value of an integral.
