Evaluate the definite integral.
$$
\int_{1}^{2}\left(\frac{12}{x^{2}}-1\right) d x=
$$

Question

Evaluate the definite integral.
$$
\int_{1}^{2}\left(\frac{12}{x^{2}}-1\right) d x=
$$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Evaluate the integral of each term separately. ‖ ‖ step 2 ⋮ The integral of $\frac{12}{x^2}$ is $-12 \cdot \frac{1}{x}$. ‖ ‖ step 3 ⋮ The integral of $-1$ is $-x$. ‖ ‖ step 4 ⋮ Combine the integrals to get the indefinite integral: $-12 \cdot \frac{1}{x} - x + C$. ‖ ‖ step 5 ⋮ Evaluate the indefinite integral from $x=1$ to $x=2$. ‖ ‖ step 6 ⋮ Plug in the upper limit: $-12 \cdot \frac{1}{2} - 2 = -6 - 2 = -8$. ‖ ‖ step 7 ⋮ Plug in the lower limit: $-12 \cdot \frac{1}{1} - 1 = -12 - 1 = -13$. ‖ ‖ step 8 ⋮ Subtract the value at the lower limit from the value at the upper limit: $(-8) - (-13) = 5$. ‖ ∻Answer∻ ⚹ 5 ⚹ ∻Key Concept∻ ⚹Integration of Rational Functions⚹ ∻Explanation∻ ⚹The integral of a rational function $\frac{A}{x^n}$ is $\frac{-A}{(n-1)x^{n-1}}$ for $n eq 1$, and the integral of a constant is the constant times the variable of integration. The definite integral is evaluated by finding the difference between the values of the indefinite integral at the upper and lower limits.⚹◊What is the value of $\int_{1}^{2}\left(\frac{12}{x^{2}}-1 ight)dx$?⍭ Generate me a similar question◊

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