Question

Math

Posted 5 months ago

```
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) $u=\sqrt{x}$
```

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Answer from Sia

Posted 5 months ago

Solution by Steps

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 $dx$.

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