Integration of other expressions

If, instead of having an expression with all terms in the form ax^n, like:

\frac{dy}{dx}=\frac13x^4-8x^{-\frac12}

we instead have something like this:

\frac{dy}{dx}=\frac1{3x^{-4}}-\frac8{\sqrt x}

…then we need to convert it into the form of the first expression.

Integrate 5x^2 + \frac7x - \frac3{\sqrt x}

• Rewrite each term in the form ax^n:
  • 5x^2 stays the same - already in the right form.
  • \frac7x can be rewritten as 7x^{-1}.
  • \frac3{\sqrt x} can be rewritten as 3x^{-\frac12}.
• So the expression becomes:
  • 5x^2 + 7x^{-1} - 3x^{-\frac12}
• Now integrate this expression term-by-term:
  • \int 5x^2 \, dx = \frac53 x^3
  • \int 7x^{-1} \, dx = 7 \ln|x|
  • \int -3x^{-\frac12} \, dx = -6x^{\frac12}
• Adding them all together, we have:
  • \int \left(5x^2 + \frac7x - \frac3{\sqrt x}\right) dx = \frac53 x^3 + 7 \ln|x| - 6x^{\frac12} + c

Find \int \left(\frac2{x^3} + \frac5{\sqrt[3]{x}} - \frac4{x^2}\right) dx

• Rewrite each term in the form ax^n:
  • \frac2{x^3} can be rewritten as 2x^{-3}.
  • \frac5{\sqrt[3]{x}} can be rewritten as 5x^{-\frac13}.
  • \frac4{x^2} can be rewritten as 4x^{-2}.
• So the expression becomes:
  • 2x^{-3} + 5x^{-\frac13} - 4x^{-2}
• Now integrate this expression term-by-term:
  • \int 2x^{-3} \, dx = -x^{-2}
  • \int 5x^{-\frac13} \, dx = \frac{15}{2} x^{\frac23}
  • \int -4x^{-2} \, dx = 4x^{-1}
• Adding them all together, we have:
  • \int \left(\frac2{x^3} + \frac5{\sqrt[3]{x}} - \frac4{x^2}\right) dx = -x^{-2} + \frac{15}{2} x^{\frac23} + 4x^{-1} + c

Find f(x) given that f'(x) = \frac{3}{x^2} - \frac{2}{\sqrt x} + 4x^3 and f(1) = 7

• Rewrite each term in the form ax^n:
  • \frac{3}{x^2} can be rewritten as 3x^{-2}.
  • \frac{2}{\sqrt x} can be rewritten as 2x^{-\frac12}.
  • 4x^3 stays the same - already in the right form.
• So the expression becomes:
  • 3x^{-2} - 2x^{-\frac12} + 4x^3
• Now integrate this expression term-by-term:
  • \int 3x^{-2} \, dx = -3x^{-1}
  • \int -2x^{-\frac12} \, dx = -4x^{\frac12}
  • \int 4x^3 \, dx = x^4
• Adding them all together, we have:
  • f(x) = -3x^{-1} - 4x^{\frac12} + x^4 + c
• Now use the initial condition f(1) = 7 to find c:
  • f(1) = -3(1)^{-1} - 4(1)^{\frac12} + (1)^4 + c = -3 - 4 + 1 + c = -6 + c
  • Set this equal to 7:
    • -6 + c = 7
    • c = 13
• Put this value of c back into the expression for f(x):
  • f(x) = -3x^{-1} - 4x^{\frac12} + x^4 + 13

Find \int \left(\frac{6}{x^4} - \frac{5}{\sqrt{x}} + \frac{2}{x}\right) dx

• Rewrite each term in the form ax^n:
  • \frac{6}{x^4} can be rewritten as 6x^{-4}.
  • \frac{5}{\sqrt{x}} can be rewritten as 5x^{-\frac12}.
  • \frac{2}{x} can be rewritten as 2x^{-1}.
• So the expression becomes:
  • 6x^{-4} - 5x^{-\frac12} + 2x^{-1}
• Now integrate this expression term-by-term:
  • \int 6x^{-4} \, dx = -2x^{-3}
  • \int -5x^{-\frac12} \, dx = -10x^{\frac12}
  • \int 2x^{-1} \, dx = 2 \ln|x|
• Adding them all together, we have:
  • \int \left(\frac{6}{x^4} - \frac{5}{\sqrt{x}} + \frac{2}{x}\right) dx = -2x^{-3} - 10x^{\frac12} + 2 \ln|x| + c

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