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# Find the integral $\int\frac{1+\sqrt{x}}{\sqrt{x}}dx$

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##  Final Answer

$2\sqrt{x}+x+C_0$
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##  Step-by-step Solution 

Problem to solve:

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

Specify the solving method

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We can solve the integral $\int\frac{1+\sqrt{x}}{\sqrt{x}}dx$ 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 $\sqrt{x}$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=\sqrt{x}$

Learn how to solve integrals of rational functions problems step by step online.

$u=\sqrt{x}$

Learn how to solve integrals of rational functions problems step by step online. Find the integral int((1+x^1/2)/(x^1/2))dx. We can solve the integral \int\frac{1+\sqrt{x}}{\sqrt{x}}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} it's a good candidate for substitution. Let's define a variable u and assign it to the choosen part. 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. Isolate dx in the previous equation. Rewriting x in terms of u.

##  Final Answer

$2\sqrt{x}+x+C_0$

##  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 int((1+x^1/2)/(x^1/2))dx using partial fractionsSolve int((1+x^1/2)/(x^1/2))dx using basic integralsSolve int((1+x^1/2)/(x^1/2))dx using u-substitutionSolve int((1+x^1/2)/(x^1/2))dx using integration by partsSolve int((1+x^1/2)/(x^1/2))dx using trigonometric substitution

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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
|◻|
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

### Main topic:

Integrals of Rational Functions

~ 0.08 s

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