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Magazine Chapter 7: Problem 7
Evaluate. $$ \int \frac{x}{\sqrt{1+x}} d x $$
Short Answer
Expert verified
\( \int \frac{x}{\sqrt{1+x}} \ dx = \frac{2(1+x)^{3/2}}{3} - 2(1+x)^{1/2} + C \)
Step by step solution
01
Set Up for Substitution
To evaluate the given integral \( \int \frac{x}{\sqrt{1+x}} \, dx \), we first use the substitution method. Let's substitute \( u = \sqrt{1+x} \), which implies \( u^2 = 1+x \) or \( x = u^2 - 1 \). Differentiating both sides with respect to \( x \), we get \( du = \frac{1}{2\sqrt{1+x}} \, dx \). Solving for \( dx \), we find \( dx = 2u \, du \).
02
Substitute in the Integral
Substitute \( x = u^2 - 1 \) and \( dx = 2u \, du \) into the integral: \( \int \frac{x}{\sqrt{1+x}} \, dx = \int \frac{u^2 - 1}{u} \cdot 2u \, du \). Simplify the expression to \( \int 2(u^2 - 1) \, du \).
03
Simplify the Integral
The integral becomes \( \int 2(u^2 - 1) \, du = \int 2u^2 \, du - \int 2 \, du \), which simplifies to \( 2 \int u^2 \, du - 2 \int 1 \, du \).
04
Integrate Each Term
Calculate each integral separately. The first integral \( \int u^2 \, du \) equals \( \frac{u^3}{3} \), and the second integral \( \int 1 \, du \) equals \( u \). So the expression becomes \( 2 \cdot \frac{u^3}{3} - 2u = \frac{2u^3}{3} - 2u \).
05
Substitute Back to Original Variable
Since \( u = \sqrt{1+x} \), substitute back: \( \frac{2(\sqrt{1+x})^3}{3} - 2\sqrt{1+x} \). Simplify it to \( \frac{2(1+x)^{3/2}}{3} - 2(1+x)^{1/2} \).
06
Add the Constant of Integration
Don鈥檛 forget the constant of integration \( C \) for indefinite integrals. Thus, the complete solution is \( \frac{2(1+x)^{3/2}}{3} - 2(1+x)^{1/2} + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
substitution method
When solving integrals, the substitution method is a powerful technique that helps to simplify the process. Imagine substitution as a way of changing variables to make integration more manageable. This method is especially useful when dealing with integrals that involve complex or composite functions.
Here's a quick summary:
Here's a quick summary:
- Choose a substitution variable, usually denoted by \( u \), to replace a complicated expression in the integral.
- Express the original variable in terms of \( u \), and find \( dx \) in terms of \( du \).
- Replace all occurrences of the original variable and \( dx \) in the integral with their \( u \) counterparts.
- Perform the integration using the new variable \( u \).
- Finally, substitute back the original variable to express the solution in terms of the original variable.
definite integral
A definite integral provides the net area under a curve between two specified points. Unlike an indefinite integral, it results in a specific numerical value. The process involves evaluating the antiderivative at the upper and lower limits of integration and subtracting the two results.
Some key points about definite integrals:
Some key points about definite integrals:
- They are evaluated over a closed interval \([a, b]\), and produce a number that represents the net signed area under the curve.
- The notation for a definite integral is \( \int_a^b f(x) \, dx \).
- When calculating, find the antiderivative of the function and then evaluate it at \( b \) and \( a \), finally subtract \( F(a) \) from \( F(b) \).
indefinite integral
The indefinite integral of a function captures a family of antiderivatives. Unlike the definite integral, it does not provide a numerical value but expresses the general form of the solutions.
Important aspects of indefinite integrals include:
Important aspects of indefinite integrals include:
- The indefinite integral is expressed as \( \int f(x) \, dx = F(x) + C \).
- It represents a whole family of functions, which vary by a constant \( C \).
- The process of finding an indefinite integral is essentially finding the antiderivative of the given function.
- Always remember to include the constant of integration \( C \), since it represents the infinite number of possible antiderivatives.
