Step 1

Given:

\(\int_{0}^{1}t^{\frac{5}{2}}(\sqrt{t}-3t)dt\)

Step 2

\(\int_{0}^{1}t^{\frac{5}{2}}(\sqrt{t}-3t)dt=\int_{0}^{1}(t^{3}-3t^{\frac{7}{2}})dt\)

\(=[\frac{t^{4}}{4}-6\frac{t^{\frac{9}{2}}}{9}]_{0}^{1}\)

\(=[\frac{t^{4}}{4}-\frac{2t^{\frac{9}{2}}}{3}]_{0}^{1}\)

\(=[\frac{(1)^{4}}{4}-\frac{2(1)^{\frac{9}{2}}}{3}]-[\frac{(0)^{4}}{4}-\frac{2(0)^{\frac{9}{2}}}{3}]\)

\(=\frac{1}{4}-\frac{2}{3}-(0)\)

\(=\frac{-5}{12}\)

Given:

\(\int_{0}^{1}t^{\frac{5}{2}}(\sqrt{t}-3t)dt\)

Step 2

\(\int_{0}^{1}t^{\frac{5}{2}}(\sqrt{t}-3t)dt=\int_{0}^{1}(t^{3}-3t^{\frac{7}{2}})dt\)

\(=[\frac{t^{4}}{4}-6\frac{t^{\frac{9}{2}}}{9}]_{0}^{1}\)

\(=[\frac{t^{4}}{4}-\frac{2t^{\frac{9}{2}}}{3}]_{0}^{1}\)

\(=[\frac{(1)^{4}}{4}-\frac{2(1)^{\frac{9}{2}}}{3}]-[\frac{(0)^{4}}{4}-\frac{2(0)^{\frac{9}{2}}}{3}]\)

\(=\frac{1}{4}-\frac{2}{3}-(0)\)

\(=\frac{-5}{12}\)