Integrating algebraic functions involving brackets and powers
When integrating algebraic expressions of the form:
\(\int {{{(ax + b)}^n}} dx \to \frac{{{{(ax + b)}^{n + 1}}}}{{a(n + 1)}} + c\)
Example
Find \(\int {{{(x + 3)}^3}}\,\,dx\)
Solution
\(\int {{{(x + 3)}^3}}\,\,dx\)
\(= \frac{{{{(x + 3)}^4}}}{{1 \times 4}} + c\)
\(= \frac{{{{(x + 3)}^4}}}{4} + c\)
Question
Find \(\int {\frac{{dx}}{{\sqrt {x - 9} }}}\)
\(\int {\frac{{dx}}{{\sqrt {x - 9} }}}\)
\(= \int {\frac{1}{{{{(x - 9)}^{\frac{1}{2}}}}}}\,\,dx\)
\(= \int {{{(x - 9)}^{ - \frac{1}{2}}}}\,\,dx\)
\(= \frac{{{{(x - 9)}^{\frac{1}{2}}}}}{{1 \times \frac{1}{2}}} + c\)
\(= 2\sqrt {x - 9} + c\)
Question
Extension
Find \(\int {{{(2x + 5)}^4}}\,\,dx\)
\(\int {{{(2x + 5)}^4}}\,\,dx\)
\(= \frac{{{{(2x + 5)}^5}}}{{2 \times 5}} + c\)
\(= \frac{{{{(2x + 5)}^5}}}{{10}} + c\)
Question
Extension
Find \(\int {\frac{1}{{\sqrt[3]{{3x + 1}}}}}\,\,dx\)
\(\int {\frac{1}{{\sqrt[3]{{3x + 1}}}}}\,\,dx\)
\(= \int {{{(3x + 1)}^{ - \frac{1}{3}}}}\,\,dx\)
\(= \frac{{{{(3x + 1)}^{\frac{2}{3}}}}}{{3 \times \frac{2}{3}}} + c\)
\(= \frac{{\sqrt[3]{{{{(3x + 1)}^2}}}}}{2} + c\)
