Integrate x+e^x from 2 to 4

\int_{2}^{4}\left(x+e^x\right)dx

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Answer

$53.2091$

Step by step solution

Problem

$\int_{2}^{4}\left(x+e^x\right)dx$
1

The integral of a sum of two or more functions is equal to the sum of their integrals

$\int_{2}^{4} e^xdx+\int_{2}^{4} xdx$
2

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\int_{2}^{4} e^xdx+\left[\frac{1}{2}x^2\right]_{2}^{4}$
3

Evaluate the definite integral

$\int_{2}^{4} e^xdx-1\cdot 2^2\cdot \frac{1}{2}+4^2\cdot \frac{1}{2}$
4

Multiply $\frac{1}{2}$ times $-1$

$\int_{2}^{4} e^xdx+2^2\left(-\frac{1}{2}\right)+4^2\cdot \frac{1}{2}$
5

Calculate the power

$\int_{2}^{4} e^xdx+4\left(-\frac{1}{2}\right)+16\cdot \frac{1}{2}$
6

Multiply $\frac{1}{2}$ times $16$

$\int_{2}^{4} e^xdx-2+8$
7

Subtract the values $8$ and $-2$

$\int_{2}^{4} e^xdx+6$
8

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$\left[e^x\right]_{2}^{4}+6$
9

Evaluate the definite integral

$6+e^2\left(-1\right)+e^4$
10

Calculate the power

$6+7.3891\left(-1\right)+54.5982$
11

Add the values $54.5982$ and $6$

$7.3891\left(-1\right)+60.5982$
12

Multiply $-1$ times $7.3891$

$60.5982-7.3891$
13

Subtract the values $60.5982$ and $-7.3891$

$53.2091$

Answer

$53.2091$

Problem Analysis

Main topic:

Integral calculus

Time to solve it:

0.22 seconds

Views:

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