Find the integral: x^ 3 -x^ 2 +x-1 x-1 d x - Vidyadip

Question

Find the integral: $\int \frac{x^{3}-x^{2}+x-1}{x-1} d x$

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Answer

I = $\int \frac{x^{3}-x^{2}+x-1}{x-1} d x$
Now the numerator can be factorized as,
x³ - x² + x - 1 = x²(x - 1) + 1(x - 1)
x³ - x² + x - 1 = (x² + 1)(x - 1)
Now putting this in given integral we get,
I = $\frac{x^{3}-x^{2}+x-1}{x-1}=\frac{\left(x^{2}+1\right)(x-1)}{x-1}=x^{2}+1$
= $\int\left(x^{2}+1\right) d x$
= $\int x^{2} d x+\int 1 . d x$
= $\frac{x^{3}}{3}+x+C$

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