Evaluate the following definite integrals: int_0^1(x^4+7e^x-3)dx

Evaluate the following definite integrals: int_0^1(x^4+7e^x-3)dx

Question
Applications of integrals
asked 2020-11-20
Evaluate the following definite integrals:
\(\displaystyle{\int_{{0}}^{{1}}}{\left({x}^{{4}}+{7}{e}^{{x}}-{3}\right)}{\left.{d}{x}\right.}\)

Answers (1)

2020-11-21
Step 1
To Evaluate the following definite integrals:
Step 2
Given That
\(\displaystyle{\int_{{0}}^{{1}}}{\left({x}^{{4}}+{7}{e}^{{x}}-{3}\right)}{\left.{d}{x}\right.}\)
\(\displaystyle={{\left[\frac{{x}^{{5}}}{{5}}+{7}{e}^{{x}}-{3}{x}\right]}_{{0}}^{{1}}}{\left[\begin{array}{c} \int{x}^{{n}}{\left.{d}{x}\right.}=\frac{{{x}^{{n}}+{1}}}{{{n}+{1}}}+{c}\\\int{e}^{{x}}{\left.{d}{x}\right.}={e}^{{x}}+{c}\end{array}\right]}\)
\(\displaystyle=\frac{{1}}{{5}}+{7}{e}^{{1}}-{3}{\left({1}\right)}-{0}-{7}{e}^{{0}}+{3}{\left({0}\right)}\)
\(\displaystyle=\frac{{1}}{{5}}+{7}{e}-{3}-{7}\)
\(\displaystyle=\frac{{1}}{{5}}+{7}{e}-{10}\)
\(\displaystyle={7}{e}-\frac{{{49}}}{{5}}\)
\(\displaystyle\therefore{\int_{{0}}^{{1}}}{\left({x}^{{4}}+{7}{e}^{{x}}-{3}\right)}{\left.{d}{x}\right.}={7}{e}-\frac{{49}}{{5}}\)
0

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