Question

Evaluate the integral

\(\) \(\int\limits_{{\rm{ - 1}}}^{\rm{1}} {{\rm{t(1 - t}}{{\rm{)}}^{\rm{2}}}{\rm{dt}}} \)\(\)

Short answer

The value of the integral is\(\)\(\left( {{\rm{ - }}\frac{{\rm{4}}}{{\rm{3}}}} \right)\)

Step-by-step solution

Expanding integrand.

\(\begin{array}{c}{\rm{t(1 - t}}{{\rm{)}}^{\rm{2}}}{\rm{ = t(1 - 2t + }}{{\rm{t}}^{\rm{2}}}{\rm{)}}\\{\rm{ = t - 2}}{{\rm{t}}^{\rm{2}}}{\rm{ + }}{{\rm{t}}^{\rm{3}}}\end{array}\)

Therefore we have to evaluate

\(\int\limits_{{\rm{ - 1}}}^{\rm{1}} {} {\rm{(t - 2}}{{\rm{t}}^{\rm{2}}}{\rm{ + }}{{\rm{t}}^{\rm{3}}}{\rm{)dt}}\)

General antiderivative.

Find the general antiderivative using the indefinite integral

And ignore (just for the moment) the boundaries of integration.

\(\begin{array}{c}\int {{\rm{(t - 2}}{{\rm{t}}^{\rm{2}}}{\rm{ + }}{{\rm{t}}^{\rm{3}}}{\rm{)dt = }}\frac{{{{\rm{t}}^{\rm{2}}}}}{{\rm{2}}}{\rm{ - 2 \times }}\frac{{{{\rm{t}}^{\rm{3}}}}}{{\rm{3}}}} {\rm{ \times }}\frac{{{{\rm{t}}^{\rm{4}}}}}{{\rm{4}}}{\rm{ + C = }}\frac{{\rm{1}}}{{\rm{2}}}{{\rm{t}}^{\rm{2}}}{\rm{ - }}\frac{{\rm{2}}}{{\rm{3}}}{{\rm{t}}^{\rm{3}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{4}}}{{\rm{t}}^{\rm{4}}}{\rm{ + C}}\\{\rm{ = F(t)}}\\\end{array}\)

We will leave out the arbitrary constant C, since it cancels out when calculating a definite integral.

Assigning the value

\(\begin{array}{c}\int\limits_{{\rm{ - 1}}}^{\rm{1}} {{\rm{(t - 2}}{{\rm{t}}^{\rm{2}}}{\rm{ + t)dt = }}\left( {\frac{{\rm{1}}}{{\rm{2}}}{{\rm{t}}^{\rm{2}}}{\rm{ - }}\frac{{\rm{2}}}{{\rm{3}}}{{\rm{t}}^{\rm{3}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{4}}}{{\rm{t}}^{\rm{4}}}} \right)_{{\rm{ - 1}}}^{\rm{1}}} {\rm{ = }}\left( {\frac{{\rm{1}}}{{\rm{2}}}{{{\rm{(1)}}}^{\rm{2}}}{\rm{ - }}\frac{{\rm{2}}}{{\rm{3}}}{{{\rm{(1)}}}^{\rm{3}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{4}}}{{{\rm{(1)}}}^{\rm{4}}}} \right){\rm{ - }}\left( {\frac{{\rm{1}}}{{\rm{2}}}{{{\rm{( - 1)}}}^{\rm{2}}}{\rm{ - }}\frac{{\rm{2}}}{{\rm{3}}}{{{\rm{( - 1)}}}^{\rm{3}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{4}}}{{{\rm{( - 1)}}}^{\rm{4}}}} \right)\\{\rm{ = }}\left( {\left| {\frac{{\rm{1}}}{{\rm{2}}}{\rm{ - }}\frac{{\rm{2}}}{{\rm{3}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{4}}}} \right|} \right){\rm{ - }}\left( {\frac{{\rm{1}}}{{\rm{2}}}{\rm{ + }}\frac{{\rm{2}}}{{\rm{3}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{4}}}} \right)\\{\rm{ = }}\left( {{\rm{ - }}\frac{{\rm{4}}}{{\rm{3}}}} \right)\end{array}\)