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Find the integral $\int\frac{x^2}{\sqrt{x^2+6}}dx$

Step-by-step Solution

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 Solution

$3\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{89}{218}u+\frac{\sqrt{u^{2}-6}}{\sqrt{6}}\right)+C_0$
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 Step-by-step Solution 

Specify the solving method

We could not solve this problem by using the method: Integrate by partial fractions

1

We can solve the integral $\int\frac{x^2}{\sqrt{x^2+6}}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $\sqrt{x^2+6}$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=\sqrt{x^2+6}$
2

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

$du=\left(x^2+6\right)^{-\frac{1}{2}}xdx$
3

Isolate $dx$ in the previous equation

$\frac{du}{\left(x^2+6\right)^{-\frac{1}{2}}x}=dx$
4

Rewriting $x$ in terms of $u$

$x=\sqrt{u^{2}-6}$
5

Substituting $u$, $dx$ and $x$ in the integral and simplify

$\int\sqrt{u^{2}-6}du$
6

We can solve the integral $\int\sqrt{u^{2}-6}du$ by applying integration method of trigonometric substitution using the substitution

$u=\sqrt{6}\sec\left(\theta \right)$
7

Now, in order to rewrite $d\theta$ 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

$du=\sqrt{6}\sec\left(\theta \right)\tan\left(\theta \right)d\theta$
8

Substituting in the original integral, we get

$\int\sqrt{6}\sqrt{6\sec\left(\theta \right)^{2}-6}\sec\left(\theta \right)\tan\left(\theta \right)d\theta$
9

Factor the polynomial $6\sec\left(\theta \right)^{2}-6$ by it's greatest common factor (GCF): $6$

$\int\sqrt{6}\sqrt{6\left(\sec\left(\theta \right)^2-1\right)}\sec\left(\theta \right)\tan\left(\theta \right)d\theta$
10

The power of a product is equal to the product of it's factors raised to the same power

$\int\frac{6}{\log^{3}\left(10\right)}\sqrt{\sec\left(\theta \right)^2-1}\sec\left(\theta \right)\tan\left(\theta \right)d\theta$
11

Apply the trigonometric identity: $\sec\left(\theta \right)^2-1$$=\tan\left(\theta \right)^2$, where $x=\theta$

$\int\frac{6}{\log^{3}\left(10\right)}\sqrt{\tan\left(\theta \right)^2}\sec\left(\theta \right)\tan\left(\theta \right)d\theta$
12

The integral of a function times a constant ($\frac{6}{\log^{3}\left(10\right)}$) is equal to the constant times the integral of the function

$\frac{6}{\log^{3}\left(10\right)}\int\sqrt{\tan\left(\theta \right)^2}\sec\left(\theta \right)\tan\left(\theta \right)d\theta$
13

Simplify $\sqrt{\tan\left(\theta \right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

$\frac{6}{\log^{3}\left(10\right)}\int\tan\left(\theta \right)\sec\left(\theta \right)\tan\left(\theta \right)d\theta$
14

When multiplying two powers that have the same base ($\tan\left(\theta \right)$), you can add the exponents

$\frac{6}{\log^{3}\left(10\right)}\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta$
15

We identify that the integral has the form $\int\tan^m(x)\sec^n(x)dx$. If $n$ is odd and $m$ is even, then we need to express everything in terms of secant, expand and integrate each function separately

$\frac{6}{\log^{3}\left(10\right)}\int\left(\sec\left(\theta \right)^2-1\right)\sec\left(\theta \right)d\theta$
16

Multiply the single term $\sec\left(\theta \right)$ by each term of the polynomial $\left(\sec\left(\theta \right)^2-1\right)$

$\frac{6}{\log^{3}\left(10\right)}\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta$
17

Expand the integral $\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\frac{6}{\log^{3}\left(10\right)}\int\sec\left(\theta \right)^{3}d\theta+\frac{6}{\log^{3}\left(10\right)}\int-\sec\left(\theta \right)d\theta$
18

The integral $\frac{6}{\log^{3}\left(10\right)}\int\sec\left(\theta \right)^{3}d\theta$ results in: $\frac{1}{2}\sqrt{x^2+6}x+3\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)$

$\frac{1}{2}\sqrt{x^2+6}x+3\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)$
19

Gather the results of all integrals

$3\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x+\frac{6}{\log^{3}\left(10\right)}\int-\sec\left(\theta \right)d\theta$
20

The integral $\frac{6}{\log^{3}\left(10\right)}\int-\sec\left(\theta \right)d\theta$ results in: $-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{89}{218}u+\frac{\sqrt{u^{2}-6}}{\sqrt{6}}\right)$

$-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{89}{218}u+\frac{\sqrt{u^{2}-6}}{\sqrt{6}}\right)$
21

Gather the results of all integrals

$3\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{89}{218}u+\frac{\sqrt{u^{2}-6}}{\sqrt{6}}\right)$
22

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

$3\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{89}{218}u+\frac{\sqrt{u^{2}-6}}{\sqrt{6}}\right)+C_0$

$3\ln\left(\frac{89}{218}\sqrt{x^2+6}+\frac{89}{218}x\right)+\frac{1}{2}\sqrt{x^2+6}x-\frac{6}{\log^{3}\left(10\right)}\ln\left(\frac{89}{218}u+\frac{\sqrt{u^{2}-6}}{\sqrt{6}}\right)+C_0$

 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^2)/((x^2+6)^0.5))dx using basic integralsSolve integral of ((x^2)/((x^2+6)^0.5))dx using u-substitutionSolve integral of ((x^2)/((x^2+6)^0.5))dx using integration by partsSolve integral of ((x^2)/((x^2+6)^0.5))dx using trigonometric substitution

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0
a
b
c
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f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
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