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int(x^(2)-4)/(2sqrtx)dx

∫x2−42xdx \int \frac{x^{2}-4}{2 \sqrt{x}} d x

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Find indefinite integrals using the substitution and by partsright chevron icon

Full solution

Q. ∫x2−42xdx \int \frac{x^{2}-4}{2 \sqrt{x}} d x
  1. Separate Terms: Simplify the integral by separating the terms in the numerator.\newline∫x2−42x dx=12∫x2x dx−12∫4x dx \int \frac{x^2 - 4}{2\sqrt{x}} \, dx = \frac{1}{2} \int \frac{x^2}{\sqrt{x}} \, dx - \frac{1}{2} \int \frac{4}{\sqrt{x}} \, dx
  2. Simplify Integral: Simplify each term in the integral.\newline12∫x2x dx=12∫x3/2 dx \frac{1}{2} \int \frac{x^2}{\sqrt{x}} \, dx = \frac{1}{2} \int x^{3/2} \, dx \newline12∫4x dx=2∫x−1/2 dx \frac{1}{2} \int \frac{4}{\sqrt{x}} \, dx = 2 \int x^{-1/2} \, dx
  3. Integrate Terms: Integrate each term.\newline12∫x3/2 dx=12⋅25x5/2=15x5/2 \frac{1}{2} \int x^{3/2} \, dx = \frac{1}{2} \cdot \frac{2}{5} x^{5/2} = \frac{1}{5} x^{5/2} \newline2∫x−1/2 dx=2⋅2x1/2=4x1/2 2 \int x^{-1/2} \, dx = 2 \cdot 2x^{1/2} = 4x^{1/2}
  4. Combine Integrated Terms: Combine the integrated terms.\newline15x5/2−4x1/2 \frac{1}{5} x^{5/2} - 4x^{1/2}

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