If f((3 x-4)/(3 x+4))=x+2, then ∫f(x) d x is equal to (A) ex- ^2 ln|(3 x-4)/(3 x+4)|+c (B) -(8)/

step 1 In this question, it is given that F of 3x minus 4 upon 3x plus 4 is equal to x plus 2. Then we

If f((3 x-4)/(3 x+4))=x+2, then ∫f(x) d x is equal to (A)...

If $f\left(\frac{3 x-4}{3 x+4}\right)=x+2$, then $\int f(x) d x$ is equal to
(A) $\operatorname{ex}-{ }^{2} \ln \left|\frac{3 x-4}{3 x+4}\right|+c$
(B) $-\frac{8}{3} \ln |x-1|+\frac{2}{3} x+c$
(C) $\frac{8}{3} \ln |x-1|+\frac{x}{3}+c$
(D) none of these

Key Concepts

Substitution in Integration

Change of variables, or substitution, is a crucial technique in integration. It involves replacing a complex expression with a simpler variable to reframe the integral into a standard form that is easier to evaluate. This method frequently appears in problems where functions are given in a transformed state, allowing one to map back to a basic integration problem.

Rational Function Integration and Logarithmic Forms

Integrating rational functions often requires breaking down the expression into simpler parts, such as through partial fraction decomposition. Many times, this process leads to integrals that yield logarithmic functions. Recognizing these patterns is an essential skill for solving integrals that involve rational expressions.

Functional Equations

A functional equation relates the values of a function at different points. In many problems, the function is defined implicitly by an equation involving transformations of its argument. Understanding how to manipulate such equations is essential for determining the explicit form of the function before applying further operations such as integration.

Inverse Functions

The inverse function method involves solving for the original variable in terms of a transformed variable. This technique is especially useful when the functional equation involves a complicated change of variables. By expressing one variable in terms of the other, one can often rewrite the function in a more familiar or accessible form.

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