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Magazine Chapter 5: Problem 35
Calculate. $$\int \frac{1}{\sqrt{x \sqrt{x}+x}} d x$$
Short Answer
Expert verified
The solution of the integral is \(2\ln|x\sqrt{x}+2x|\).
Step by step solution
01
Substitute
Use the substitution method and let \(u = x\sqrt{x} + x\). Hence, \(u' = \frac{3}{2}x^{\frac{1}{2}} + 1\) and then \(du = (\frac{3}{2}x^{\frac{1}{2}} + 1)dx\), therefore \(dx = \frac{du}{\frac{3}{2}x^{\frac{1}{2}} + 1}\). With this, the integral becomes \(\int\frac{du}{\sqrt{u}(\frac{3}{2}x^{\frac{1}{2}} + 1)}\).
02
Simplify
Take \(\frac{3}{2}x^{\frac{1}{2}} + 1\) out of the integral: \( \int\frac{du}{u}\) * \(\frac{1}{\frac{3}{2}x^{\frac{1}{2}} + 1}\). In case of \(u = x\sqrt{x} + x\), \(x=\frac{u}{\sqrt{u}+1}\). Replace \(x\) in integral and get \(\int\frac{1}{u\sqrt{u}+1} du\).
03
Solve the Integral
The integral form is now significantly simpler and can be solved using the standard method for logarithmic integrals. Hence, the solution of integral \(\int\frac{1}{u\sqrt{u}+1} du\) is \(2\ln|u+\sqrt{u}|\).
04
Undo Substitution
Finally, replace \(u\) with \(x\sqrt{x} + x\). The solution of the original integral problem is \(2\ln|x\sqrt{x}+2x|\).
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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 valuable technique in integral calculus. It's used to simplify integrals by changing variables. In our exercise, we use substitution to make the problem solvable. The idea is to replace a part of the integrand with a simpler variable, where the derivative of this new variable appears elsewhere in the integrand.
Here's how substitution works in this context:
Here's how substitution works in this context:
- You identify a part of the integrand that can be substituted. In this example, we let \( u = x\sqrt{x} + x \).
- Then, determine \( du \) by differentiating \( u \) with respect to \( x \). This gives \( du = (\frac{3}{2}x^{\frac{1}{2}} + 1)dx \).
- We rearrange to express \( dx \) in terms of \( du \), resulting in \( dx = \frac{du}{\frac{3}{2}x^{\frac{1}{2}} + 1} \).
- Substitute these into the original integral to replace \( x \)-based expressions with \( u \), making the integral simpler.
Logarithmic Integrals
Logarithmic integrals appear often in calculus when dealing with functions of the form \( \int \frac{1}{u} \, du \). These are integrals that evaluate to a natural logarithm function. In this specific case, after simplification, we achieve an expression that closely matches the form of a logarithmic integral.
Steps in dealing with logarithmic integrals involve:
Steps in dealing with logarithmic integrals involve:
- Recognizing the integral of \( \frac{1}{u} \) which results in \( \ln|u| \).
- Adapting the expressions into this form through mathematical manipulation, as shown in the solution where the integral simplifies to \( \int \frac{1}{u\sqrt{u}+1} \, du \).
- Solving the resultant integral using standard logarithmic properties, generally yielding solutions involving the natural logarithm function.
Integral Simplification
Integral simplification is about transforming a complex integral into a simpler form that is easier to evaluate. It's an essential skill to solve integrals efficiently in calculus.
Key aspects of simplifying integrals include:
Key aspects of simplifying integrals include:
- Identifying parts of the integral that can be expressed in a more straightforward form. Recognizing patterns or common integration forms is vital.
- Using substitutions like the method above can convert cumbersome expressions into manageable ones.
- Applying algebraic simplifications either before or after substitution to make the integral more tractable.
- In this example, we worked to remove complex components by rewriting them in terms of \( u \), leading to a simplified evaluation process with the integral \( \int \frac{1}{u\sqrt{u}+1} \, du \).
