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Integration

Question 7 of 11

Beginning Integration — The Reverse of Differentiation

Question

Integrate \(\int \dfrac{3}{x^{\frac{1}{2}}}\,dx\)

Show solution Hide solution Fully worked — 5 steps
  1. Rewrite the integrand

    Express \(\frac{1}{x^{\frac{1}{2}}}\) in the form \(x^n\)

    \[ \int \frac{3}{x^{\frac{1}{2}}}\,dx = \int 3x^{-\frac{1}{2}}\,dx \]
  2. Apply the rule

    Use \(a\int x^n\,dx = \frac{ax^{n+1}}{n+1} + c\), substitute \(n = -\frac{1}{2}\).

    \[ = \frac{3x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1} + c \]
  3. Simplify the exponent and denominator
    \[ = \frac{3x^{\frac{1}{2}}}{\frac{1}{2}} + c \]
  4. Divide

    Dividing 3 by \(\frac{1}{2}\).

    \[ = 6x^{\frac{1}{2}} + c \]
  5. Write the answer as a radical

    Express as a radical.

    \[ \int \frac{3}{x^{\frac{1}{2}}}\,dx = 6\sqrt{x} + c \]

    That is, the integral is 6 times the square root of \(x\), plus the constant of integration \(c\).