Question

Evaluate the following definite integrals \int_{0}^{1}(x^{4}+7e^{x}-3)dx

Applications of integrals
ANSWERED
asked 2020-12-17
Evaluate the following definite integrals
\(\displaystyle{\int_{{{0}}}^{{{1}}}}{\left({x}^{{{4}}}+{7}{e}^{{{x}}}-{3}\right)}{\left.{d}{x}\right.}\)

Answers (1)

2020-12-18

Given that
\(\displaystyle{\int_{{{0}}}^{{{1}}}}{\left({x}^{{{4}}}+{7}{e}^{{{x}}}-{3}\right)}{\left.{d}{x}\right.}\)
\(=[\frac{x^{5}}{5}+7e^{x}-3x]_{0}^{1}\ \begin{bmatrix}\because \int x^{n}dx=\frac{x^{n+1}}{n+1}+C \\\int e^{x}dx=e^{x}+C \end{bmatrix}\)
\(\displaystyle={\frac{{{1}}}{{{5}}}}+{7}{e}^{{{1}}}-{3}{\left({1}\right)}-{0}-{7}{e}^{{{0}}}+{3}{\left({0}\right)}\)
\(\displaystyle={\frac{{{1}}}{{{5}}}}+{7}{e}\cdot-{10}\)
\(\displaystyle={7}{e}-{\frac{{{49}}}{{{5}}}}\)
\(\displaystyle\therefore{\int_{{{0}}}^{{{1}}}}{\left({x}^{{{4}}}+{7}{e}^{{{x}}}-{3}\right)}{\left.{d}{x}\right.}={7}{e}-{\frac{{{49}}}{{{5}}}}\)

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