Let f defined for all x ≠0 such that 2 f(x)+f(-x)=(1)/(x) sin(x-(1)/(x)) . Then, ∫1 / 6^e f(x) d

step 1 In this problem of integral calculus, we have given that let F define for all, so let F be the f

Let f defined for all x ≠0 such that 2 f(x)+f(-x)=(1)/(x)...

Let $\mathrm{f}$ defined for all $\mathrm{x} \neq 0$ such that $2 \mathrm{f}(\mathrm{x})+\mathrm{f}(-\mathrm{x})=\frac{1}{\mathrm{x}} \sin \left(\mathrm{x}-\frac{1}{\mathrm{x}}\right) .$ Then, $\int_{1 / 6}^{e} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$ is (a) $2 \mathrm{e}+1$ (b) $e+1$ (c) $2 \mathrm{e}-1$ (d) 0

Let $\mathrm{f}$ defined for all $\mathrm{x} \neq 0$ such that $2 \mathrm{f}(\mathrm{x})+\mathrm{f}(-\mathrm{x})=\frac{1}{\mathrm{x}} \sin \left(\mathrm{x}-\frac{1}{\mathrm{x}}\right) .$ Then, $\int_{1 / 6}^{e} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$ is
(a) $2 \mathrm{e}+1$
(b) $e+1$
(c) $2 \mathrm{e}-1$
(d) 0

Key Concepts

Substitution Methods in Integration

Substitution is a widely used method for simplifying integrals by changing the variable of integration. In contexts where the integrand includes functions evaluated at both x and -x, a substitution (such as u = -x) helps in comparing and potentially combining integrals. This step is important in deducing cancellations or reductions that lead to the final answer.

Functional Equations

This concept involves solving equations where an unknown function appears with different arguments. In this case, the relation 2f(x) + f(-x) = (1/x)sin(x - 1/x) sets up a functional equation linking the values of the function at x and -x. The process typically involves creating a system by also considering the equation with x replaced by -x and then combining the equations to solve for f(x) explicitly.

Symmetry in Functions

Analyzing symmetry properties, especially the even and odd nature of functions, plays a crucial role in this problem. The functional equation relates f(x) and f(-x), suggesting that investigating whether f exhibits even, odd, or neither symmetry can simplify computations. Recognizing these properties can lead to cancellations or simplifications in the integration process.

Integration Techniques

Integration techniques are essential when evaluating an integral that involves a function defined by a functional equation. Techniques such as splitting the integral, applying substitution, and using symmetry properties can simplify the evaluation of integrals. These methods are particularly useful when the integrand is expressed in terms of a function whose direct antiderivative is hard to find.

Transcript

Please give Ace some feedback