Question

Evaluate the indefinite integral.

\(\int {{{{\rm{(1 - 2x)}}}^{\rm{9}}}} {\rm{dx}}\).

Short answer

The solution of given integral\(\int {{{(1 - 2x)}^9}} dx\) is \( - \frac{1}{{20}} \cdot {(1 - 2x)^{10}} + C\).

Step-by-step solution

Substitution method of evaluating integral

Substitution method is a method which transforms given integral into a simple form of integral by substituting the independent variable by others.

Applying substitution method 

Let \(u = 1 - 2x\)

\(\begin{aligned}{c}du &= - 2dx\\ - \frac{{du}}{2} &= dx\end{aligned}\)

Substituting u in given function to get required results-

\(\begin{aligned}{c}\int {{{(1 - 2x)}^9}} dx &= \int {{u^9}\left( { - \frac{{du}}{2}} \right)} \\ &= - \frac{1}{2}\int {{u^9}du} \\ &= - \frac{1}{2}\left( {\frac{{{u^{9 + 1}}}}{{9 + 1}}} \right) + C\\ &= - \frac{1}{2}\left( {\frac{{{u^{10}}}}{{10}}} \right) + C\\ &= - \frac{1}{{20}} \cdot {u^{10}} + C\end{aligned}\)

Replace value of u with original value

\(\)\(\int {{{(1 - 2x)}^9}} dx = - \frac{1}{{20}} \cdot {(1 - 2x)^{10}} + C\)

Therefore, the solution of given integral\(\int {{{(1 - 2x)}^9}} dx\) is \( - \frac{1}{{20}} \cdot {(1 - 2x)^{10}} + C\).