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Solve the integral of logarithmic functions $\int\left(\frac{1}{x^2+8}+x\ln\left(x\right)\right)dx$

Step-by-step Solution

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Final Answer

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

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Expand the integral $\int\left(\frac{1}{x^2+8}+x\ln\left(x\right)\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\int\frac{1}{x^2+8}dx+\int x\ln\left(x\right)dx$

Learn how to solve integrals involving logarithmic functions problems step by step online.

$\int\frac{1}{x^2+8}dx+\int x\ln\left(x\right)dx$

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Learn how to solve integrals involving logarithmic functions problems step by step online. Solve the integral of logarithmic functions int(1/(x^2+8)+xln(x))dx. Expand the integral \int\left(\frac{1}{x^2+8}+x\ln\left(x\right)\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{1}{x^2+8}dx results in: \frac{\sqrt{2}}{4}\arctan\left(\frac{\sqrt{2}}{4}x\right). The integral \int x\ln\left(x\right)dx results in: \frac{1}{2}x^2\ln\left(x\right)-\frac{1}{4}x^2. Gather the results of all integrals.

Final Answer

$\frac{\sqrt{2}}{4}\arctan\left(\frac{\sqrt{2}}{4}x\right)-\frac{1}{4}x^2+\frac{1}{2}x^2\ln\left(x\right)+C_0$

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 (1/(x^2+8)+xlnx)dx using basic integralsSolve integral of (1/(x^2+8)+xlnx)dx using u-substitutionSolve integral of (1/(x^2+8)+xlnx)dx using integration by partsSolve integral of (1/(x^2+8)+xlnx)dx using tabular integration

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Function Plot

Plotting: $\frac{\sqrt{2}}{4}\arctan\left(\frac{\sqrt{2}}{4}x\right)-\frac{1}{4}x^2+\frac{1}{2}x^2\ln\left(x\right)+C_0$

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7
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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
|◻|
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: Integrals involving Logarithmic Functions

They are those integrals where the function that we are integrating is composed only of combinations of logarithmic functions.

Used Formulas

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