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int((8)/(sqrtx)+8sqrtx)dx

∫(8x+8x)dx \int\left(\frac{8}{\sqrt{x}}+8 \sqrt{x}\right) d x

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Find indefinite integrals using the substitution and by partsright chevron icon

Full solution

Q. ∫(8x+8x)dx \int\left(\frac{8}{\sqrt{x}}+8 \sqrt{x}\right) d x
  1. Split and Rewrite Integral: Rewrite the integral by splitting the terms.\newline\int(\(8/\sqrt{x} + 88\sqrt{x})\,dx = \int 88x^{(−1-1/22)}\,dx + \int 88x^{(11/22)}\,dx
  2. Integrate Each Term: Integrate each term separately.\newline∫8x−12 dx=8∫x−12 dx\int 8x^{-\frac{1}{2}}\,dx = 8\int x^{-\frac{1}{2}}\,dx and ∫8x12 dx=8∫x12 dx\int 8x^{\frac{1}{2}}\,dx = 8\int x^{\frac{1}{2}}\,dx
  3. Apply Power Rule: Apply the power rule for integration to each term.\newline8∫x−12dx=8×(x−12+112+1)8\int x^{-\frac{1}{2}}dx = 8 \times \left(\frac{x^{-\frac{1}{2} + 1}}{\frac{1}{2} + 1}\right) and 8∫x12dx=8×(x12+112+1)8\int x^{\frac{1}{2}}dx = 8 \times \left(\frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}\right)
  4. Simplify Exponents and Fractions: Simplify the exponents and fractions. 8×(x1/23/2)+8×(x3/23/2)8 \times \left(\frac{x^{1/2}}{3/2}\right) + 8 \times \left(\frac{x^{3/2}}{3/2}\right)
  5. Multiply by Reciprocal: Multiply through by the reciprocal of the fractions.\newline8×(23)x12+8×(23)x328 \times \left(\frac{2}{3}\right)x^{\frac{1}{2}} + 8 \times \left(\frac{2}{3}\right)x^{\frac{3}{2}}
  6. Simplify Constants: Simplify the constants. \newline163x12+163x32+C\frac{16}{3}x^{\frac{1}{2}} + \frac{16}{3}x^{\frac{3}{2}} + C
  7. Combine Terms: Combine the terms to write the final answer. final_answer=163x12+163x32+Cfinal\_answer=\frac{16}{3}x^{\frac{1}{2}} + \frac{16}{3}x^{\frac{3}{2}} + C

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