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Integrate the function $\left(x+1\right)\left(x-4\right)$

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Solution

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Final answer to the problem

$\frac{1}{2}x^2\left(x+1\right)-4x\left(x+1\right)+2x^2-\frac{1}{6}x^{3}+C_0$
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Step-by-step Solution

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1

Find the integral

$\int\left(x+1\right)\left(x-4\right)dx$
2

We can solve the integral $\int\left(x+1\right)\left(x-4\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
3

First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=\left(x+1\right)}\\ \displaystyle{du=dx}\end{matrix}$
4

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=\left(x-4\right)dx}\\ \displaystyle{\int dv=\int \left(x-4\right)dx}\end{matrix}$
5

Solve the integral

$v=\int\left(x-4\right)dx$
6

The integral of a constant is equal to the constant times the integral's variable

$\int xdx-4x$
7

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

$\frac{1}{2}x^2-4x$
8

Now replace the values of $u$, $du$ and $v$ in the last formula

$\left(\frac{1}{2}x^2-4x\right)\left(x+1\right)-\int\frac{1}{2}x^2dx-\int-4xdx$
9

Multiply the single term $x+1$ by each term of the polynomial $\left(\frac{1}{2}x^2-4x\right)$

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

The integral $-\frac{1}{2}\int x^2dx+4\int xdx$ results in: $-\frac{1}{6}x^{3}+2x^2$

$-\frac{1}{6}x^{3}+2x^2$
11

Gather the results of all integrals

$\frac{1}{2}x^2\left(x+1\right)-4x\left(x+1\right)+2x^2-\frac{1}{6}x^{3}$
12

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{1}{2}x^2\left(x+1\right)-4x\left(x+1\right)+2x^2-\frac{1}{6}x^{3}+C_0$

Final answer to the problem

$\frac{1}{2}x^2\left(x+1\right)-4x\left(x+1\right)+2x^2-\frac{1}{6}x^{3}+C_0$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Solve integral of (x+1)(x-4)dx using partial fractionsSolve integral of (x+1)(x-4)dx using basic integralsSolve integral of (x+1)(x-4)dx using u-substitutionIntegrate using trigonometric identitiesSolve integral of (x+1)(x-4)dx using trigonometric substitution

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Function Plot

Plotting: $\frac{1}{2}x^2\left(x+1\right)-4x\left(x+1\right)+2x^2-\frac{1}{6}x^{3}+C_0$

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◻2
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√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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Main Topic: Integral Calculus

Integration assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

Used Formulas

See formulas (8)

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