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

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

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

### Relevant Questions

Evaluate the integrals.
$$\displaystyle{\int_{{-{{2}}}}^{{2}}}{\left({3}{x}^{{4}}-{2}{x}+{1}\right)}{\left.{d}{x}\right.}$$
Find all rational zeros of the polynomial, and write the polynomial in factored form.
$$\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}-{3}{x}^{{{2}}}-{2}{x}+{3}$$
Is the algebraic expression a polynomial? If it is, write the polynomial in standard form:
$$\displaystyle{2}{x}+{3}{x}^{{{2}}}-{5}$$
Evaluate the following definite integrals
$$\displaystyle{\int_{{{0}}}^{{{1}}}}{\left({x}^{{{4}}}+{7}{e}^{{{x}}}-{3}\right)}{\left.{d}{x}\right.}$$
Evaluate the following integrals.
$$\displaystyle\int{\left({2}{x}^{{{3}}}-{x}^{{{2}}}+{3}{x}-{7}\right)}{\left.{d}{x}\right.}$$
Evaluate the following integrals.
$$\displaystyle{\int_{{-{2}}}^{{-{1}}}}\sqrt{{-{4}{x}-{x}^{{{2}}}}}{\left.{d}{x}\right.}$$
Evaluate the following definite integrals
$$\displaystyle{\int_{{{0}}}^{{{1}}}}{x}{e}^{{{\left(-{x}^{{{2}}}+{2}\right)}}}{\left.{d}{x}\right.}$$
Show that the differential forms in the integrals are exact. Then evaluate the integrals.
$$\displaystyle{\int_{{{\left({1},{1},{2}\right)}}}^{{{\left({3},{5},{0}\right)}}}}{y}{z}{\left.{d}{x}\right.}+{x}{z}{\left.{d}{y}\right.}+{x}{y}{\left.{d}{z}\right.}$$
Which of the following integrals are improper integrals?
1.$$\displaystyle{\int_{{{0}}}^{{{3}}}}{\left({3}-{x}\right)}^{{{2}}}{\left\lbrace{3}\right\rbrace}{\left.{d}{x}\right.}$$
2.$$\displaystyle{\int_{{{1}}}^{{{16}}}}{\frac{{{e}^{{\sqrt{{{x}}}}}}}{{\sqrt{{{x}}}}}}{\left.{d}{x}\right.}$$
3.$$\displaystyle{\int_{{{1}}}^{{\propto}}}{\frac{{{3}}}{{\sqrt{{{3}}}{\left\lbrace{x}\right\rbrace}}}}{\left.{d}{x}\right.}$$
4.$$\displaystyle{\int_{{-{2}}}^{{{2}}}}{3}{\left({x}+{1}\right)}^{{-{1}}}{\left.{d}{x}\right.}$$
a) 1 only
b)1 and 2
c)3 only
d)2 and 3
e)1,3 and 4
f)All of the integrals are improper
$$\displaystyle{\int_{{-{{2}}}}^{{-{{1}}}}}\sqrt{{-{4}{x}-{x}^{{2}}}}{\left.{d}{x}\right.}$$