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Magazine Chapter 5: Problem 31
Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. \(\int e^{x^{2}+2 x+5}(x+1) d x\)
Short Answer
Expert verified
\(\int e^{x^2+2x+5}(x+1)dx = \frac{1}{2}e^{x^2+2x+5} + C\).
Step by step solution
01
Identify the Function for Substitution
First, we notice that the expression in the exponent \(x^2 + 2x + 5\) is a quadratic polynomial. Let's see if it resembles any common substitution formulas. We can consider substituting \(u = x^2 + 2x + 5\) because its derivative, \(du\), might simplify the expression.
02
Differentiate the Substitution Variable
Differentiate the substitution variable \(u\) with respect to \(x\). This gives us:\[du = (2x + 2) \, dx\]
03
Simplify the Integral with Substitution
For the original integral \(\int e^{x^2 + 2x + 5}(x+1) \, dx\), we observe that \((x+1)\) can be expressed in terms of \(u\) as follows:When \(du = (2x + 2) \, dx\), we can rewrite this as \(du = 2(x+1)\, dx\), which implies that \((x+1)\, dx = \frac{1}{2}du\).
04
Substitute in the Integral
Using the substitution, the integral becomes:\[ \int e^u \cdot \frac{1}{2} du \]
05
Integrate in Terms of u
Now, integrate \(\int \frac{1}{2} e^u \, du\). This can be calculated as:\[\frac{1}{2} \int e^u \, du = \frac{1}{2} e^u + C\] where \(C\) is the constant of integration.
06
Substitute Back the Original Variable
Since \(u = x^2 + 2x + 5\), replace \(u\) back with the expression in terms of \(x\). The final answer is:\[\frac{1}{2} e^{x^2 + 2x + 5} + C\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a powerful tool in integral calculus. It simplifies complex integrals by transforming them into a more manageable form. Essentially, this method involves substituting a part of the integrand with a new variable, often denoted as \( u \). The goal is to convert the given integral into a basic form that is easier to evaluate.
To use the substitution method effectively, follow these steps:
To use the substitution method effectively, follow these steps:
- Identify a portion of the integrand that can be replaced by a new variable \( u \).
- Differentiate this new variable \( u \) to find \( du \).
- Adjust the integral accordingly to incorporate \( du \).
- Integrate the new expression.
- Substitute back the original variable to express the solution in terms of the original integrand.
Quadratic Polynomial
A quadratic polynomial is a polynomial of degree two. It takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In the context of integration, quadratic polynomials often arise in the exponents of exponential functions.
For integration problems, recognizing and handling quadratic polynomials within the integrand is crucial:
For integration problems, recognizing and handling quadratic polynomials within the integrand is crucial:
- They can often be the perfect candidates for substitution.
- Rewriting expressions using these polynomials can simplify the derivative calculations.
- When substituting, ensure that the differential \( du \) aligns well with the remaining parts of the integrand.
Integral Calculus
Integral calculus is a branch of mathematics concerned with finding integrals. These represent the area under a curve or the accumulation of quantities. The process of finding an integral is called integration.
There are different types of integrals:
There are different types of integrals:
- Indefinite Integrals: These do not have set limits and include a constant of integration \( C \). The task is to find a general form of the antiderivative.
- Definite Integrals: These involve limits and give a numerical value representing area under the curve between specified points.
Exponential Function
An exponential function is a mathematical function of the form \( e^x \), where the base \( e \) is Euler's number, approximately equal to 2.71828. When dealing with integrals involving exponential functions, recognizing the structure of the integrand is crucial.
Here鈥檚 why exponential functions are important in calculus:
Here鈥檚 why exponential functions are important in calculus:
- They exhibit continuous growth, modeling various natural phenomena like population growth and radioactive decay.
- Integrals involving exponential functions often relate to exponential growth or decay problems.
- When the exponential function includes a complicated exponent, such as a quadratic polynomial, substitution often aids in simplifying the integral.
