Evaluate the integrals. \int_{-2}^{2}(3x^{4}-2x+1)dx

Evaluate the integrals. \int_{-2}^{2}(3x^{4}-2x+1)dx

Question
Applications of integrals
asked 2021-01-28
Evaluate the integrals.
\(\displaystyle{\int_{{-{2}}}^{{{2}}}}{\left({3}{x}^{{{4}}}-{2}{x}+{1}\right)}{\left.{d}{x}\right.}\)

Answers (1)

2021-01-29
Step 1: Given that
Evaluate integrals.
\(\displaystyle{\int_{{-{2}}}^{{{2}}}}{\left({3}{x}^{{{4}}}-{2}{x}+{1}\right)}{\left.{d}{x}\right.}\)
Step 2: Solving the integral
We have,
PSK\int_{-2}^{2}(3x^{4}-2x+1)dx=\int_{2}^{2}3x^{4}dx- 2\int_{-2}^{2}xdx+\int_{-2}^{2}1dxZSK
\(\displaystyle={3}{{\left[{\frac{{{x}^{{{5}}}}}{{{5}}}}\right]}_{{-{2}}}^{{{2}}}}-{2}{{\left[{\frac{{{x}^{{{2}}}}}{{{2}}}}\right]}_{{-{2}}}^{{{2}}}}+{{\left[{x}\right]}_{{-{2}}}^{{{2}}}}\)
\(\displaystyle={\frac{{{3}}}{{{5}}}}{\left[{2}^{{{5}}}-{\left(-{2}\right)}^{{{5}}}\right]}-{\left[{2}^{{{2}}}-{\left(-{2}\right)}^{{{2}}}\right]}+{\left[{2}-{\left(-{2}\right)}\right]}\)
\(\displaystyle={\frac{{{3}}}{{{5}}}}{\left({32}-{\left(-{32}\right)}\right)}-{\left({4}-{4}\right)}+{\left({2}+{2}\right)}\)
\(\displaystyle={\frac{{{3}}}{{{5}}}}{\left({64}\right)}-{0}+{4}\)
\(\displaystyle={\frac{{{192}}}{{{5}}}}+{4}\)
\(\displaystyle={\frac{{{192}+{20}}}{{{5}}}}\)
\(\displaystyle={\frac{{{212}}}{{{5}}}}\)
=42.4
0

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