Problem 153
Question
$$ y=-\frac{x}{1+8 x^{3}}+\frac{1}{12} \ln \frac{(1+2 x)^{2}}{1-2 x+4 x^{2}}+\frac{\sqrt{3}}{6} \tan ^{-1} \frac{4 x-1}{\sqrt{3}} $$
Step-by-Step Solution
Verified Answer
The simplified function is:
\(y = -\frac{x}{1+8x^3} + \frac{1}{12} \ln{\frac{1+2x}{1+2x-4}} + \frac{\sqrt{3}}{6}\tan^{-1}\frac{4x-1}{\sqrt{3}}\)
1Step 1: Simplify the Rational Function
The first term is \(-\frac{x}{1+8x^3}\). The term cannot be simplified further, and therefore this term remains the same.
2Step 2: Simplify the Natural Logarithmic Function
The second term is \(\frac{1}{12}\ln\frac{(1+2x)^2}{1-2x+4x^2}\). Let's simplify the expression inside the natural logarithm.
We have,
\(1-2x+4x^2 = (1+2x)^2 - 4(1+2x)\)
The expression inside the natural logarithm becomes:
\(\frac{(1+2x)^2}{(1+2x)^2 - 4(1+2x)}\)
Now, let's let \(u = 1+2x\). The expression inside the natural logarithm can be rewritten as:
\(\frac{u^2}{u^2-4u}=\frac{u}{u-4}\)
Now we can simplify the natural logarithm term:
\(\frac{1}{12}\ln\frac{(1+2x)^2}{1-2x+4x^2} = \frac{1}{12} \ln{\frac{u}{u-4}}\)
However, since we want the expression in terms of x, we reverse the substitution by replacing u with \(1+2x\):
\(\frac{1}{12} \ln{\frac{u}{u-4}} = \frac{1}{12} \ln{\frac{1+2x}{1+2x-4}}\)
Hence, the second term simplifies to:
\(\frac{1}{12} \ln{\frac{1+2x}{1+2x-4}}\)
3Step 3: Simplify the Arctangent Function
The third term is \(\frac{\sqrt{3}}{6}\tan^{-1}\frac{4x-1}{\sqrt{3}}\). This term cannot be simplified further, and therefore this term remains the same.
4Step 4: Combine the Simplified Terms
Now that we have simplified each term, we can combine them to get the simplified function:
\(y = -\frac{x}{1+8x^3} + \frac{1}{12} \ln{\frac{1+2x}{1+2x-4}} + \frac{\sqrt{3}}{6}\tan^{-1}\frac{4x-1}{\sqrt{3}}\)
Key Concepts
Simplifying Rational FunctionsNatural Logarithmic Function SimplificationInverse Trigonometric Functions
Simplifying Rational Functions
When handling complex equations with several terms, starting by simplifying rational functions can make them more manageable. A rational function is a fraction of two polynomials. Simplifying these functions often involves reducing the fraction to its lowest terms, factorizing numerators and denominators, and canceling out common factors.
For example, in the exercise, the term \( -\frac{x}{1+8x^3} \) is already in its simplest form, as the numerator and denominator share no common factors. If there had been common factors, we would have divided them out to simplify the function further. It's strategic to scrutinize each term for potential simplifications before attempting to solve or combine them with other terms in the equation.
For example, in the exercise, the term \( -\frac{x}{1+8x^3} \) is already in its simplest form, as the numerator and denominator share no common factors. If there had been common factors, we would have divided them out to simplify the function further. It's strategic to scrutinize each term for potential simplifications before attempting to solve or combine them with other terms in the equation.
Natural Logarithmic Function Simplification
The natural logarithm, denoted as \( \ln(x) \), is the inverse of the exponential function \( e^x \) and it's pivotal in calculus. Simplifying natural logarithmic functions often involves using properties of logarithms, such as the product, quotient, and power rules.
In the provided exercise, we simplified the logarithmic function by first expanding and then reducing the inside expression. We used a clever substitution by letting \( u = 1+2x \) to condense the function, which allowed us to cancel out common terms and thus simplify the logarithm. Such substitutions are instrumental for simplification, as they can turn a seemingly complex expression into a more straightforward one. Finally, we replaced \( u \) with its expression in terms of \( x \) to obtain the simplified logarithmic function in the original variable.
In the provided exercise, we simplified the logarithmic function by first expanding and then reducing the inside expression. We used a clever substitution by letting \( u = 1+2x \) to condense the function, which allowed us to cancel out common terms and thus simplify the logarithm. Such substitutions are instrumental for simplification, as they can turn a seemingly complex expression into a more straightforward one. Finally, we replaced \( u \) with its expression in terms of \( x \) to obtain the simplified logarithmic function in the original variable.
Inverse Trigonometric Functions
Inverse trigonometric functions, like the arctangent \( \tan^{-1} \), are used to determine the angle that produces a given trigonometric value. These functions can sometimes be simplified by using geometric properties or identities, but there are cases where they are at their simplest form and cannot be condensed further.
In the original function, \( y \) includes the term \( \frac{\sqrt{3}}{6}\tan^{-1}\frac{4x-1}{\sqrt{3}} \) which is already in its simplest form. This term represents the angle whose tangent is \( \frac{4x-1}{\sqrt{3}} \). Although not simplified further in this problem, it's important to recognize when inverse trigonometric functions are in their basic state and when they can be manipulated using trigonometric identities to simplify or integrate them into a larger problem-solving strategy.
In the original function, \( y \) includes the term \( \frac{\sqrt{3}}{6}\tan^{-1}\frac{4x-1}{\sqrt{3}} \) which is already in its simplest form. This term represents the angle whose tangent is \( \frac{4x-1}{\sqrt{3}} \). Although not simplified further in this problem, it's important to recognize when inverse trigonometric functions are in their basic state and when they can be manipulated using trigonometric identities to simplify or integrate them into a larger problem-solving strategy.
Other exercises in this chapter
Problem 151
$$ y=\ln \sqrt[4]{\frac{x^{2}+x+1}{x^{2}-x+1}}+\frac{1}{2 \sqrt{3}}\left(\tan ^{-1} \frac{2 x+1}{\sqrt{3}}+\tan ^{-1} \frac{2 x-1}{\sqrt{3}}\right) $$
View solution Problem 152
$$ \left.y=\cos ^{-1} \frac{x^{2 n}-1}{x^{2 n}+1} \text { \\{ Ans. }-\frac{2 n x^{n-1}}{x^{2 n}+1} \text { if } n \text { is even and }-\frac{2 n x^{n}}{|x|\lef
View solution Problem 156
$$ y=(\sin x)^{\cos x} $$
View solution Problem 157
$$ y=(\tan 2 x)^{\cot ^{\frac{x}{2}}} $$
View solution