Math Insight

The simplest integration substitutions


The simplest kind of chain rule application $${d\over dx}f(ax+b)=a\cdot f'(x)$$ (for constants $a,b$) can easily be run backwards to obtain the corresponding integral formulas: some and illustrative important examples are \begin{align*} \int \cos(ax+b)\;dx &={1\over a}\cdot \sin(ax+b)+C\\ \int e^{ax+b}\;dx &= {1\over a}\cdot e^{ax+b}+C\\ \int \sqrt{ax+b}\;dx &= {1\over a}\cdot { (ax+b)^{3/2} \over 3/2 }+C\\ \int {1\over ax+b}\;dx &= {1\over a}\cdot \ln (ax+b)+C \end{align*}

Putting numbers in instead of letters, we have examples like \begin{align*} \int \cos(3x+2)\;dx&={1\over 3}\cdot \sin(3x+2)+C\\ \int e^{4x+3}\;dx&={1\over 4}\cdot e^{4x+3}+C\\ \int \sqrt{-5x+1}\;dx&={1\over -5}\cdot { (-5x+1)^{3/2} \over 3/2 }+C\\ \int {1\over 7x-2}\;dx&={1\over 7}\cdot \ln (7x-2)+C \end{align*}

Since this kind of substitution is pretty undramatic, and a person should be able to do such things by reflex rather than having to think about it very much.


  1. $\int e^{3x+2}\; dx=?$
  2. $\int \cos(2-5x)\; dx=?$
  3. $\int \sqrt{3x-7}\; dx=?$
  4. $\int \sec^2({2x+1})\; dx=?$
  5. $\int (5x^7+e^{6-2x}+23+{ 2 \over x })\,dx=?$
  6. $\int \cos(7-11x)\,dx=?$