Heba was asked to find this integral using $u$-substitution:
$$
\int(-12 x-1) \sqrt{-6 x^{2}-x+1} d x
$$

How should Heba define $u$ ?

Choose 1 answer:
(A) $u=-6 x^{2}-x+1$
(B) $u=\sqrt{-6 x^{2}-x+1}$
(c) $u=-12 x-1$
(D) $u=\sqrt{x}$

Question

Heba was asked to find this integral using $u$-substitution:
$$
\int(-12 x-1) \sqrt{-6 x^{2}-x+1} d x
$$

How should Heba define $u$ ?

Choose 1 answer:
(A) $u=-6 x^{2}-x+1$
(B) $u=\sqrt{-6 x^{2}-x+1}$
(c) $u=-12 x-1$
(D) $u=\sqrt{x}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the integral using $u$-substitution, we look for a function inside the integral that when differentiated, gives a result similar to the other terms in the integral. ‖ ‖ step 2 ⋮ We identify the function under the square root, $-6x^2 - x + 1$, as a potential $u$ because its derivative, $-12x - 1$, is present outside the square root. ‖ ‖ step 3 ⋮ Define $u$ as the function under the square root: $u = -6x^2 - x + 1$. ‖ ‖ step 4 ⋮ Differentiate $u$ with respect to $x$ to find $du$: $du = (-12x - 1)dx$. This shows that our choice of $u$ is correct because $(-12x - 1)dx$ is present in the integral. ‖ ‖ step 5 ⋮ Substitute $u$ and $du$ into the integral and solve. The integral becomes $\int \sqrt{u} \, du$, which is straightforward to integrate. ‖ ***A*** ∻Key Concept∻ ⚹$u$-substitution⚹ ∻Explanation∻ ⚹In $u$-substitution, we choose $u$ to be a function inside the integral that simplifies the integral when we substitute $du$ for the corresponding terms in $dx$.⚹◊What is the correct definition of $u$ in the context of $u$-substitution for the integral $\int(-12x-1)\sqrt{-6x^{2}-x+1}dx$?⍭ Generate me a similar question◊

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