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

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

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

### Relevant Questions

Evaluate each of the following integrals.
$$\int_{0}^{2}(x^{2}+2x-3)^{3}(4x+4)dx$$
Evaluate each of the following integrals.
$$\int\frac{e^{x}}{1+e^{x}}dx$$
Evaluate the ff, improper integrals.
$$\int_{1}^{\infty}\frac{1}{x^{3}}dx$$
Evaluate the following integral: $$\int \frac{x+3}{x-1}dx$$
Evaluate the following definite integrals
$$\displaystyle{\int_{{{0}}}^{{{1}}}}{\left({x}^{{{4}}}+{7}{e}^{{{x}}}-{3}\right)}{\left.{d}{x}\right.}$$
Evaluate the ff, improper integrals.
$$\int_{-2}^{\infty}\sin x dx$$
$$\int_{-1}^{1}f(x)dx$$
Find the indefinite integral $$\int \ln(\frac{x}{3})dx$$ (a) using a table of integrals and (b) using the Integration by parts method.
Given $$\int_{2}^{5}f(x)dx=17$$ and $$\int_{2}^{5}g(x)dx=-2$$, evaluate the following.
(a)$$\int_{2}^{5}[f(x)+g(x)]dx$$
(b)$$\int_{2}^{5}[g(x)-f(x)]dx$$
(c)$$\int_{2}^{5}2g(x)dx$$
(d)$$\int_{2}^{5}3f(x)dx$$