Examples of General Power of Integration

Example: Evaluate the integral $$\int {\left( {2x + 7} \right){{\left( {{x^2} + 7x + 3} \right)}^{\frac{4}{5}}}dx} $$ with respect to $$x$$

We have integral \[I = \int {\left( {2x + 7} \right){{\left( {{x^2} + 7x + 3} \right)}^{\frac{4}{5}}}dx} \]

Here $$f\left( x \right) = {x^2} + 7x + 3$$ implies that $$f’\left( x \right) = 2x + 7$$

We observe that the derivation of the given function is in the given problem, so using the general power formula of integration, we have
\[\begin{gathered}\int {\left( {2x + 7} \right){{\left( {{x^2} + 7x + 3} \right)}^{\frac{4}{5}}}dx} = \frac{{{{\left( {{x^2} + 7x + 3} \right)}^{\frac{4}{5} + 1}}}}{{\frac{4}{5} + 1}} + c \\ \Rightarrow \int {\left( {2x + 7} \right){{\left( {{x^2} + 7x + 3} \right)}^{\frac{4}{5}}}dx} = \frac{{{{\left( {{x^2} + 7x + 3} \right)}^{\frac{9}{5}}}}}{{\frac{9}{5}}} + c \\ \Rightarrow \int {\left( {2x + 7} \right){{\left( {{x^2} + 7x + 3} \right)}^{\frac{4}{5}}}dx} = \frac{5}{9}{\left( {{x^2} + 7x + 3} \right)^{\frac{9}{5}}} + c \\ \end{gathered} \]

Example: Integrate $${\sin ^6}x\cos x$$ with respect to $$x$$.

Consider the function to be integrated \[I = \int {{{\sin }^6}x\cos xdx} \]

Here $$f\left( x \right) = \sin x$$ implies that $$f’\left( x \right) = \cos x$$

We observe that the derivation of the given function is in the given problem, so using the general power formula of integration, we have
\[\begin{gathered} \int {{{\sin }^6}x\cos xdx} = \frac{{{{\left( {\sin x} \right)}^{6 + 1}}}}{{6 + 1}} + c \\ \Rightarrow \int {{{\sin }^6}x\cos xdx} = \frac{1}{7}{\sin ^7}x + c \\ \end{gathered} \]