Problem 36
Question
Evaluate the indefinite integral. \(\int \frac{x}{1+x^{4}} d x\)
Step-by-Step Solution
Verified Answer
\( \frac{1}{2} \arctan(x^2) + C \)
1Step 1: Recognize the Formula
The given integral is \( \int \frac{x}{1+x^{4}} \mathrm{d}x \). Notice that the integrand allows consideration of a substitution because the derivative of \(x^4\) is \(4x^3\), which has a similar form to the denominator.
2Step 2: Perform a Suitable Substitution
Choose a substitution to simplify the integrand. Let \( u = 1 + x^4 \). Then, compute \( \frac{du}{dx} = 4x^3 \), which implies \( du = 4x^3 \, dx \), but adjust it to suit the integrand by rewriting it as \( du = 4x^3 \, dx \equiv x \, dx = \frac{1}{4x^2} du \). In this case, a simpler substitution is \( v = x^2 \), leading to \( dv = 2x \, dx \), or \( x \, dx = \frac{1}{2} \, dv \).
3Step 3: Change Variables in the Integral
Using \( v = x^2 \), substitute into the integral: it transforms the integral to \( \frac{1}{2}\int \frac{1}{v^2 + 1} \, dv \). This matches the standard integral formula for the arctangent function.
4Step 4: Evaluate the Integral
The integral \( \frac{1}{2} \int \frac{1}{v^2 + 1} \, dv \) is \( \frac{1}{2} \arctan(v) + C \), where \( C \) is the constant of integration.
5Step 5: Back-Substitute to Original Variable
Replace \( v \) with \( x^2 \) to convert the result back to context of the original variable: \( \frac{1}{2} \arctan(x^2) + C \).
6Step 6: State the Final Solution
Thus, the indefinite integral \( \int \frac{x}{1+x^{4}} \, dx \) is \( \frac{1}{2} \arctan(x^2) + C \).
Key Concepts
Substitution MethodIndefinite IntegralArctangent Function
Substitution Method
The substitution method is a powerful technique in calculus that simplifies the process of finding integrals. This method involves replacing a complex part of the integrand with a single variable, making the integral easier to evaluate. In the given exercise, the substitution method is used to transform the integral \( \int \frac{x}{1+x^{4}} \, dx \) into a simpler form.
To begin, identify a substitution that will simplify the fractional expression. Here, we chose \( v = x^2 \) because it results in \( dv = 2x \, dx \). The expression \( x \, dx \) can then be rewritten as \( \frac{1}{2} \ dv \). This substitution effectively changes the integral into a form that resembles standard integral formulas.
There are crucial elements in using the substitution method:
To begin, identify a substitution that will simplify the fractional expression. Here, we chose \( v = x^2 \) because it results in \( dv = 2x \, dx \). The expression \( x \, dx \) can then be rewritten as \( \frac{1}{2} \ dv \). This substitution effectively changes the integral into a form that resembles standard integral formulas.
There are crucial elements in using the substitution method:
- Choose a substitution \( u = g(x) \) that simplifies the integrand.
- Express \( dx \) in terms of \( du \).
- Change the limits of integration if you are working with a definite integral.
- Transform the integral into one that can be easily solved.
Indefinite Integral
In calculus, an indefinite integral represents a family of functions whose derivative is the integrand. It's usually denoted without specific limits and includes a constant of integration \( C \) to embody the infinite number of antiderivatives.
The given problem focuses on evaluating the indefinite integral \( \int \frac{x}{1+x^{4}} \, dx \). This involves finding a general solution without any specific boundary values. Unlike definite integrals which compute a numerical value over an interval, an indefinite integral results in a function expression.
Key features of indefinite integrals include:
The given problem focuses on evaluating the indefinite integral \( \int \frac{x}{1+x^{4}} \, dx \). This involves finding a general solution without any specific boundary values. Unlike definite integrals which compute a numerical value over an interval, an indefinite integral results in a function expression.
Key features of indefinite integrals include:
- They do not have upper and lower bounds.
- The result always ends with "+ \( C \)", representing the constant term.
- They capture all possible antiderivatives of the integrand.
Arctangent Function
The arctangent function, often expressed as \( \arctan(x) \), is an inverse of the tangent function used frequently in calculus, especially in integration. This function is pivotal when dealing with integrands that have expressions of the form \( \frac{1}{x^2 + 1} \).
In this exercise, after employing the substitution \( v = x^2 \), the integral resolves into \( \int \frac{1}{v^2 + 1} \, dv \). This expression precisely aligns with the standard integral formula for the arctangent function, which is \( \int \frac{1}{1+u^2} \, du = \arctan(u) + C \). So, \( \int \frac{1}{v^2 + 1} \, dv = \arctan(v) + C \).
Highlights of the arctangent function:
In this exercise, after employing the substitution \( v = x^2 \), the integral resolves into \( \int \frac{1}{v^2 + 1} \, dv \). This expression precisely aligns with the standard integral formula for the arctangent function, which is \( \int \frac{1}{1+u^2} \, du = \arctan(u) + C \). So, \( \int \frac{1}{v^2 + 1} \, dv = \arctan(v) + C \).
Highlights of the arctangent function:
- It is the inverse function of the tangent, yielding the angle whose tangent is the given number.
- Useful in integration where the integrand matches the format \( \frac{1}{x^2 + 1} \).
- The derivative of \( \arctan(x) \) is \( \frac{1}{1 + x^2} \).
Other exercises in this chapter
Problem 36
Verify by differentiation that the formula is correct. \(\int x \cos x d x=x \sin x+\cos x+C\)
View solution Problem 36
Given that $$\int_{0}^{1} 3 x \sqrt{x^{2}+4} d x=5 \sqrt{5}-8$$, what is $$\int_{1}^{0} 3 u \sqrt{u^{2}+4} d u ?$$
View solution Problem 37
\(37-38\) Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen. \(\int\left(\cos x+\frac{1}{2} x\right)
View solution Problem 37
Suppose that \(f(1)=2, f(4)=7, f^{\prime}(1)=5, f^{\prime}(4)=3,\) and \(f^{\prime \prime}\) is continuous. Find the value of \(\int_{1}^{4} x f^{\prime \prime}
View solution