Integral of \( \displaystyle \sqrt{2 x - 1} \)

Problem 4.654 · medium

1. \[ \int \sqrt{2 x - 1}\, dx \]
   integral algebraStart with the integral of the function. Rewrite the square root as a fractional exponent.✓ Proved
2. \[ = \int \sqrt{2} \sqrt{x - \frac{1}{2}}\, dx \]
   algebraFactor out the constant 2 from the expression inside the power.✓ Proved
3. \[ = \sqrt{2} \int \sqrt{x - \frac{1}{2}}\, dx \]
   linearityMove the constant coefficient outside the integral.✓ Proved
4. \[ = \frac{2 \sqrt{2} \left(x - \frac{1}{2}\right)^{\frac{3}{2}}}{3} \]
   antiderivative algebra algebraApply the power rule for integration. Simplify the fraction 1/(3/2) to 2/3. Rewrite (x - 1/2) as (2*x - 1)/2.✓ Proved
5. \[ = \frac{\left(2 x - 1\right)^{\frac{3}{2}}}{3} \]
   algebra algebra algebra simplifyDistribute the exponent 3/2 to the numerator and denominator. Group the powers of 2 in the denominator. Simplify the exponent subtraction 3/2 - 1/2 = 1. Simplify the coefficient 2/3 / 2 to 1/3.✓ Proved