Indefinite Integration Question - None

Question

Assertion (A): When $f(x) = \frac{x^2 + 1}{2}$, $\int \frac{dx}{x} = 2\ln|x| + c$

Reason (R): $\int (h(x))^n h'(x) dx = \frac{(h(x))^{n+1}}{n+1} + C$

Accepted Answer

Step 1: The standard integral $\int \frac{dx}{x} = \ln|x| + c$ is independent of the definition of $f(x)$

Step 2: If $f(x) = \frac{x^2+1}{2}$, this doesn't change the integral of $\frac{1}{x}$

Step 3: Therefore, $\int \frac{dx}{x} = \ln|x| + c$, not $2\ln|x| + c$

Step 4: The power rule for composition $(h(x))^n h'(x) dx = \frac{(h(x))^{n+1}}{n+1} + C$ is correct

∴ A is false but R is true.

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