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Magazine Chapter 5: Problem 36
Find or evaluate the integral. (Complete the square, if necessary.) $$ \int \frac{2}{\sqrt{-x^{2}+4 x}} d x $$
Short Answer
Expert verified
The solution for \int \frac{2}{\sqrt{-x^{2}+4 x}} d x is 2 ln(|\sqrt{1+(x-1)^{2}}+(x-1)|)+C.
Step by step solution
01
Rewrite the square inside the square root.
The term -\x^{2}+ 4x can be written as perfect square by splitting the 4x into 2x and 2x and adding and subtracting the square of the co-efficient of the second 2x i.e 1. So, it results in -\((x^{2} - 2x + 1) -1\). Now this equation can be simplified as -\((x-1)^{2} -1\).
02
Use Trigonometric Substitution.
Now based on the term inside the square root, select a trigonometric function to replace \(x - 1\). A suitable choice is the sine function due to its derivative cos function. So, use a substitution where \(x - 1 = sin(胃)\). Next step is to substitute this into our integral.
03
Substitute and Simplify the Integral.
Substitute \(x - 1 = sin(胃)\) into the integral. As a result we have \int \frac{2}{\sqrt{1 - sin^{2}(胃)}} cos(胃) d胃. The square root can be simplified using Pythagorean identity, resulting in \int 2 sec(胃) cos(胃) d胃. This simplifies to \int 2 sec(胃) d胃.
04
Solve the Integral.
Our exercise now is to solve 2 \int sec(胃) d胃. The integral of sec(胃) can be found directly on standard tables, which is ln(|sec(胃) + tan(胃)|). Therefore, our integral evaluates to 2 ln(|sec(胃) + tan(胃)|) + C.
05
Back-substitute the original variable.
Since we initially substituted \(x - 1 = sin(胃)\), we need to replace 胃 back in terms of x. So, in the end, the final integral is 2 ln(|sec(胃) + tan(胃)|) + C = 2 ln(|\sqrt{1+(x-1)^{2}} +(x-1) |)+C.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a technique used to simplify expressions, especially useful when dealing with quadratic expressions. It involves turning a quadratic expression into a perfect square trinomial so it can be easily factored or integrated.
For the expression inside our integral, \(-x^{2} + 4x\), we reframe it to make it easier to work with.
For the expression inside our integral, \(-x^{2} + 4x\), we reframe it to make it easier to work with.
- First, reorganize \-x^{2} + 4x\ as \(- (x^{2} - 4x)\).
- Next, divide the coefficient of \-4x\ by 2 and square it: \( (2)^{2} = 4 \).
- Add and subtract this square inside the expression: \(-(x^{2} - 4x + 4 - 4)\).
- Rewrite it as \(-((x-2)^{2} - 4)\).
Trigonometric Substitution
Trigonometric substitution is a clever method to tackle integrals involving square roots. By substituting a trigonometric expression, we simplify complex root expressions.
In our example, we use trigonometric substitution for the expression including a square root derived through completing the square.
In our example, we use trigonometric substitution for the expression including a square root derived through completing the square.
- Substitute \(x-1 = \sin(\( \theta \))\). This choice leverages the identity \(\sin^{2}(\theta) + \cos^{2}(\theta) = 1\).
- Transform the integral under this substitution, allowing simplification of the square root.
Definite Integrals
Definite integrals compute the net area under a curve between given limits. They provide a numerical value, which represents the accumulated total across an interval when plotted on a graph.
- While our example is an indefinite integral, understanding definite integrals provides context to the entire function behavior across specific intervals.
- The solved integral offers antiderivatives, which, within defined limits, evaluates the integral across specific bounds.
Indefinite Integrals
An indefinite integral represents a family of functions, each differing by a constant, and thus includes a \(C\) where the exact function depends on initial conditions.
This reflects the solution鈥檚 inherent flexibility when not bounded by limits.
This reflects the solution鈥檚 inherent flexibility when not bounded by limits.
- In the problem at hand, the integral \( \int \frac{2}{\sqrt{-x^{2}+4x}} \, dx \) demonstrates a classic indefinite integral, showcasing solving techniques like completing the square and trigonometric substitution.
- The \(C\) added to the solution indicates all potential shifted solutions along the vertical axis on a graph.
