QuestionFind the integral of $\frac{1}{x^{2}-x+1}$ with respect to $x$ .

Studdy Solution

STEP 1

Question: Which of the following is the completed square form of $x^{2}-x+1$ ?
A) $\left(x-\frac{1}{2}\right)^{2} + \frac{1}{4}$ B) $\left(x-\frac{1}{2}\right)^{2} + \frac{3}{4}$ C) $\left(x+\frac{1}{2}\right)^{2} + \frac{3}{4}$ D) $\left(x+\frac{1}{2}\right)^{2} + \frac{1}{4}$
Answer: B) $\left(x-\frac{1}{2}\right)^{2} + \frac{3}{4}$

STEP 2

Question: If we let $u = x - \frac{1}{2}$ , what will be the new integral?
A) $\int \frac{1}{u^{2} + \frac{3}{4}} d u$ B) $\int \frac{1}{u^{2} + \frac{1}{2}} d u$ C) $\int \frac{1}{u^{2} + 1} d u$ D) $\int \frac{1}{u^{2} + 2} d u$
Answer: A) $\int \frac{1}{u^{2} + \frac{3}{4}} d u$

STEP 3

Question: How can we simplify the integral $\int \frac{1}{u^{2} + \frac{3}{4}} d u$ further by multiplying the numerator and denominator by a certain number?
A) $\int \frac{4}{3} \cdot \frac{1}{\frac{4}{3}u^{2} + 1} d u$ B) $\int \frac{4}{3} \cdot \frac{1}{\frac{3}{4}u^{2} + 1} d u$ C) $\int \frac{3}{4} \cdot \frac{1}{\frac{4}{3}u^{2} + 1} d u$ D) $\int \frac{3}{4} \cdot \frac{1}{\frac{3}{4}u^{2} + 1} d u$
Answer: A) $\int \frac{4}{3} \cdot \frac{1}{\frac{4}{3}u^{2} + 1} d u$

STEP 4

Question: What is the result of integrating the expression $\int \frac{4}{3} \cdot \frac{1}{\left(\frac{2}{\sqrt{3}}\right)^{2} + u^{2}} d u$ using the standard integral formula?
A) $\frac{4}{3} \cdot \frac{\sqrt{3}}{2} \arctan\left(\frac{u \sqrt{3}}{2}\right) + C$ B) $\frac{4\sqrt{3}}{3} \arctan\left(\frac{u \sqrt{3}}{2}\right) + C$ C) $\frac{2}{3} \cdot \frac{\sqrt{3}}{2} \arctan\left(\frac{u \sqrt{3}}{2}\right) + C$ D) $\frac{2\sqrt{3}}{3} \arctan\left(\frac{u \sqrt{3}}{2}\right) + C$
Answer: D) $\frac{2\sqrt{3}}{3} \arctan\left(\frac{u \sqrt{3}}{2}\right) + C$

STEP 5

Question: After substituting back $u$ in terms of $x$ , what is the final result of the integral $\int \frac{1}{x^{2}-x+1} d x$ ?
A) $\frac{2\sqrt{3}}{3} \arctan\left(\frac{(x-\frac{1}{2}) \sqrt{3}}{2}\right) + C$ B) $\frac{2}{3} \arctan\left(\frac{(x-\frac{1}{2}) \sqrt{3}}{2}\right) + C$ C) $\frac{2\sqrt{3}}{3} \arctan\left(\frac{(x+\frac{1}{2}) \sqrt{3}}{2}\right) + C$ D) $\frac{2}{3} \arctan\left(\frac{(x+\frac{1}{2}) \sqrt{3}}{2}\right) + C$
Answer: A) $\frac{2\sqrt{3}}{3} \arctan\left(\frac{(x-\frac{1}{2}) \sqrt{3}}{2}\right) + C$

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