$Find $\int\frac{dx}{2x^2-2x+1}$$

Question

$Find $\int\frac{dx}{2x^2-2x+1}$$

Please helm me integrate the function

$$\int\frac{dx}{2x^2-2x+1}.$$

Calculus Integrals

Lilian00

36

Report

Answer

The answer is accepted.

Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.

Answer 1

Answers can only be viewed under the following conditions:

The questioner was satisfied with and accepted the answer, or
The answer was evaluated as being 100% correct by the judge.

View the answer

We can solve this by rewriting the function and completing the square
\[ \int\frac{dx}{2x^2-2x+1}=\int \frac{dx}{2(x^2-x+\frac{1}{4})+\frac{1}{2}}=\int \frac{dx}{2(x-\frac{1}{2})^2+\frac{1}{2}}\]
\[=\frac{1}{2}\int \frac{dx}{(x-\frac{1}{2})^2+\frac{1}{4}}=\frac{1}{2}\arctan (\frac{x-\frac{1}{2}}{\frac{1}{2}})+c=\frac{1}{2}\times \frac{1}{2}\arctan (2x-1)+c\]
\[=\frac{1}{4}\arctan (2x-1)+c.\]
Here we have used the antiderivative formula
\[\int \frac{dx}{(x-b)^2+a^2}=\frac{1}{a} \arctan (\frac{x-a}{a})+c.\]

Erdos

4.7K

$Find $\int\frac{dx}{2x^2-2x+1}$$

Answer

Related Questions

Search