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}