Heba was asked to find this integral using -substitution:

How should Heba define ?

Ask by Schofield Lowe. in the United States

Jan 22,2025

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 $$ u $$ ?

UpStudy AI · Accepted Answer

To solve the integral using $$ u $$-substitution, it's effective to choose $$ u $$ as the expression inside the square root. This choice simplifies the integral because the derivative of $$ u $$ will match the coefficient in front of the square root.

Let's define:
$$
u = -6x^2 - x + 1
$$

Now, compute the derivative of $$ u $$ with respect to $$ x $$:
$$
\frac{du}{dx} = -12x - 1
$$
$$
du = (-12x - 1) \, dx
$$

This matches the coefficient $$(-12x - 1)$$ in the original integral, allowing the integral to be rewritten in terms of $$ u $$.

**Final Answer:**
Choose u to equal –6 x² minus x plus one. In symbols,

u = −6 x² − x + 1
Heba should define $$ u = -6x^2 - x + 1 $$.

Heba was asked to find this integral using -substitution:

How should Heba define ?

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Heba was asked to find this integral using -substitution: How should Heba define ?