# Integrate 4x^3+2/7x from a to b

## \int_{a}^{b}\left(4x^3+\frac{2}{7}\cdot x\right)dx

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## Step by step solution

Problem

$\int_{a}^{b}\left(4x^3+\frac{2}{7}\cdot x\right)dx$
1

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

$\int_{a}^{b}\frac{2}{7}xdx+\int_{a}^{b}4x^3dx$
2

Taking the constant out of the integral

$\frac{2}{7}\int_{a}^{b} xdx+\int_{a}^{b}4x^3dx$
3

Taking the constant out of the integral

$\frac{2}{7}\int_{a}^{b} xdx+4\int_{a}^{b} x^3dx$
4

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

$\frac{2}{7}\int_{a}^{b} xdx+\left[4\frac{x^{4}}{4}\right]_{a}^{b}$
5

Simplify the fraction

$\frac{2}{7}\int_{a}^{b} xdx+\left[x^{4}\right]_{a}^{b}$
6

Evaluate the definite integral

$\frac{2}{7}\int_{a}^{b} xdx-x^{4}+x^{4}$
7

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

$\frac{2}{7}\left[\frac{1}{2}x^2\right]_{a}^{b}-x^{4}+x^{4}$
8

Evaluate the definite integral

$\frac{2}{7}\left(\frac{1}{2}x^2-1\cdot \frac{1}{2}x^2\right)-x^{4}+x^{4}$
9

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

$\frac{2}{7}\left(\frac{1}{2}x^2-\frac{1}{2}x^2\right)-x^{4}+x^{4}$
10

Adding $\frac{1}{2}x^2$ and $-\frac{1}{2}x^2$

$\frac{2}{7}\cdot 0x^2-x^{4}+x^{4}$
11

Any expression multiplied by $0$ is equal to $0$

$0-x^{4}+x^{4}$
12

$x+0=x$, where $x$ is any expression

$x^{4}-x^{4}$
13

Subtracting $x^{4}$ and $x^{4}$

$0$

$0$

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### Main topic:

Integral calculus

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