Question

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.∫ $$(x^2 + 6x$$) / $$(x^2$$ + $$3)^2 dx$$

Accepted Answer

Identify a substitution that simplifies the integral. Notice that the denominator is $$ (x^2 + 3)^2 $$ and the numerator contains terms $$ x^2 + 6x . $$Consider letting $$ u = x^2 + 3 $$ because its derivative relates to the numerator. Compute the derivative of $$ u $$ with respect to $$ x $$: $$ \frac{du}{dx} = 2x $$, which implies $$ du = 2x \, dx . $$This will help express parts of the integral in terms of $$ u $$ and $$ du . $$Rewrite the integral in terms of $$ u $$ and $$ du . $$Express $$ x^2 + 6x  in $$terms of $$ u $$ and $$ x $$, and replace $$ dx $$ using $$ du = 2x \, dx  or$$ $$ dx = \frac{du}{2x} . $$This step may require splitting the integral or manipulating the numerator to match the substitution. After substitution, simplify the integral to a form involving $$ u $$ only, ideally matching a standard integral from the table, such as $$ \int \frac{du}{u^2}  or $$similar. Integrate the simplified expression with respect to $$ u $$, then substitute back $$ u = x^2 + 3  to $$express the answer in terms of $$ x .$$

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ (x^2 + 6x) / (x^2 + 3)^2 dx

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Key Concepts

Integration by Substitution

Recognizing Derivatives within the Integral

Using Integral Tables

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.∫ (x^2 + 6x) / (x^2 + 3)^2 dx