# Matrices Calculator

## Get detailed solutions to your math problems with our Matrices step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here!

Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

### Difficult Problems

1

Solved example of gaussian elimination

$\int\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}dx$
2

Rewrite the fraction $\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}$ in $3$ simpler fractions using partial fraction decomposition

$\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}=\frac{A}{x+2}+\frac{B}{\left(x+1\right)^2}+\frac{C}{x+1}$
3

Find the values for the unknown coefficients: $A, B, C$. The first step is to multiply both sides of the equation from the previous step by $\left(x+2\right)\left(x+1\right)^2$

$5x^2+14x+10=\left(x+2\right)\left(x+1\right)^2\left(\frac{A}{x+2}+\frac{B}{\left(x+1\right)^2}+\frac{C}{x+1}\right)$
4

Multiplying polynomials

$5x^2+14x+10=\frac{A\left(x+2\right)\left(x+1\right)^2}{x+2}+\frac{B\left(x+2\right)\left(x+1\right)^2}{\left(x+1\right)^2}+\frac{C\left(x+2\right)\left(x+1\right)^2}{x+1}$
5

Simplifying

$5x^2+14x+10=A\left(x+1\right)^2+B\left(x+2\right)+C\left(x+2\right)\left(x+1\right)$
6

Expand the polynomial

$5x^2+14x+10=A\left(x+1\right)^2+Bx+2B+x^2C+2xC+Cx+2C$
7

Assigning values to $x$ we obtain the following system of equations

$\begin{matrix}2=A&\:\:\:\:\:\:\:(x=-2) \\ 58=12C+4B+9A&\:\:\:\:\:\:\:(x=2) \\ 1=B&\:\:\:\:\:\:\:(x=-1)\end{matrix}$
8

Proceed to solve the system of linear equations

$\begin{matrix}1A & + & 0B & + & 0C & =2 \\ 9A & + & 4B & + & 12C & =58 \\ 0A & + & 1B & + & 0C & =1\end{matrix}$
9

Rewrite as a coefficient matrix

$\left(\begin{matrix}1 & 0 & 0 & 2 \\ 9 & 4 & 12 & 58 \\ 0 & 1 & 0 & 1\end{matrix}\right)$
10

Reducing the original matrix to a identity matrix using Gaussian Elimination

$\left(\begin{matrix}1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 3\end{matrix}\right)$
11

The integral of $\frac{5x^2+14x+10}{\left(x+2\right)\left(x+1\right)^2}$ in decomposed fraction equals

$\int\left(\frac{2}{x+2}+\frac{1}{\left(x+1\right)^2}+\frac{3}{x+1}\right)dx$

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

$\int\frac{2}{x+2}dx+\int\left(\frac{1}{\left(x+1\right)^2}+\frac{3}{x+1}\right)dx$

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

$\int\frac{2}{x+2}dx+\int\frac{1}{\left(x+1\right)^2}dx+\int\frac{3}{x+1}dx$
12

Simplifying

$\int\frac{2}{x+2}dx+\int\frac{1}{\left(x+1\right)^2}dx+\int\frac{3}{x+1}dx$

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$2\int\frac{1}{2+x}dx$

Apply the formula: $\int\frac{n}{x+b}dx$$=n\ln\left|x+b\right|, where b=2 and n=1 2\ln\left|x+2\right| 13 The integral \int\frac{2}{x+2}dx results in: 2\ln\left|x+2\right| 2\ln\left|x+2\right| We can solve the integral \int\frac{1}{\left(x+1\right)^2}dx by applying integration by substitution method (also called U-Substitution). First, we must identify a part of the integral with a new variable, which when substituted makes the integral easier. We see that x+1 it's a good candidate for substitution. Let's define a variable u and assign it to the choosen part u=x+1 Now, in order to rewrite dx in terms of du, we need to find the derivative of u. We need to calculate du, we can do that by deriving the equation above du=dx Substituting u and dx in the integral and simplify 2\ln\left|x+2\right|+\int\frac{1}{u^2}du+\int\frac{3}{x+1}dx Rewrite the exponent using the power rule \frac{a^m}{a^n}=a^{m-n}, where in this case m=0 \int u^{-2}du Apply the power rule for integration, \displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}, where n represents a constant function -u^{-1} Substitute u back with the value that we assigned to it: x+1 -\left(x+1\right)^{-1} Applying the property of exponents, \displaystyle a^{-n}=\frac{1}{a^n}, where n is a number \frac{-1}{x+1} 14 The integral \int\frac{1}{\left(x+1\right)^2}dx results in: \frac{-1}{x+1} \frac{-1}{x+1} The integral of a constant by a function is equal to the constant multiplied by the integral of the function 3\int\frac{1}{1+x}dx Apply the formula: \int\frac{n}{x+b}dx$$=n\ln\left|x+b\right|$, where $b=1$ and $n=1$

$3\ln\left|x+1\right|$
15

The integral $\int\frac{3}{x+1}dx$ results in: $3\ln\left|x+1\right|$

$3\ln\left|x+1\right|$
16

Gather the results of all integrals

$2\ln\left|x+2\right|+\frac{-1}{x+1}+3\ln\left|x+1\right|$
17

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

$2\ln\left|x+2\right|+\frac{-1}{x+1}+3\ln\left|x+1\right|+C_0$

$2\ln\left|x+2\right|+\frac{-1}{x+1}+3\ln\left|x+1\right|+C_0$