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int(1)/(26 xsqrtx)dx

126xxdx \int \frac{1}{26 x \sqrt{x}} d x

Full solution

Q. 126xxdx \int \frac{1}{26 x \sqrt{x}} d x
  1. Rewrite with x1/2x^{-1/2}: Rewrite the integral with xx to the power of 1/2-1/2. \newline126xxdx=126x3/2dx\int\frac{1}{26x\sqrt{x}}dx = \int\frac{1}{26x^{3/2}}dx
  2. Pull out constant: Pull out the constant 126\frac{1}{26}. \newline126x32dx=(126)x32dx\int \frac{1}{26x^{\frac{3}{2}}}dx = \left(\frac{1}{26}\right)\int x^{-\frac{3}{2}}dx
  3. Integrate x32x^{-\frac{3}{2}}: Integrate x32x^{-\frac{3}{2}} with respect to xx.(126)x32dx=(126)(21)x12+C\left(\frac{1}{26}\right)\int x^{-\frac{3}{2}}\,dx = \left(\frac{1}{26}\right)\left(-\frac{2}{1}\right)x^{-\frac{1}{2}} + C
  4. Simplify expression: Simplify the expression.\newline(\frac{\(1\)}{\(26\)})(\frac{\(-2\)}{\(1\)})x^{(-\frac{\(1\)}{\(2\)})} + C = (\frac{\(-2\)}{\(26\)})x^{(-\frac{\(1\)}{\(2\)})} + C
  5. Reduce fraction: Reduce the fraction \(-\frac{2}{26} to 113-\frac{1}{13}.\left(-\frac{\(2\)}{\(26\)}\right)x^{-\frac{\(1\)}{\(2\)}} + C = \left(-\frac{\(1\)}{\(13\)}\right)x^{-\frac{\(1\)}{\(2\)}} + C
  6. Rewrite as \(\frac{1}{\sqrt{x}}: Rewrite x12x^{-\frac{1}{2}} as 1x\frac{1}{\sqrt{x}}.
    $\left(-\frac{\(1\)}{\(13\)}\right)x^{-\frac{\(1\)}{\(2\)}} + C = \left(-\frac{\(1\)}{\(13\)}\right)\left(\frac{\(1\)}{\sqrt{x}}\right) + C