Indefinite Integral Calculation
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4. $$\int \frac{x^2 + 1}{(x^3 + 3x - 5)^3} dx =$$ (A) $-\frac{3}{2} \cdot \frac{1}{(3x^2 + 3)^2} + C$ (B) $-\frac{1}{6} \cdot \frac{1}{(3x^2 + 3)^2} + C$ (C) $-\frac{3}{2} \cdot \frac{1}{(x^3 + 3x - 5)^2} + C$ (D) $-\frac{1}{6} \cdot \frac{1}{(x^3 + 3x - 5)^2} + C$
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Step by Step Written Solution
Hi Yasemin, let's solve this integration problem together step by step.
We are asked to find the integral of x squared plus one, over x cubed plus three x minus five, all raised to the third power.
Notice that the numerator looks like it might be related to the derivative of the expression inside the parentheses in the denominator. This suggests we should use u-substitution.
Method: u-Substitution
Let's set u equal to the expression inside the denominator's parentheses.
Choosing our u
Now, let's find the derivative of u with respect to x.
We can factor out a three from the expression on the right to make it clearer.
If we rearrange this, we can solve for d x. Specifically, d u over three equals x squared plus one times d x.
Now, let's substitute these values back into our original integral.
Substitution
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