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Find the integral $\int\frac{x^2-1}{\sqrt{2x-1}}dx$

Step-by-step Solution

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Solution

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Final Answer

$-\frac{3}{4}\sqrt{2x-1}+\frac{1}{6}\sqrt{\left(2x-1\right)^{3}}+\frac{1}{20}\sqrt{\left(2x-1\right)^{5}}+C_0$
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Step-by-step Solution

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Expand the fraction $\frac{x^2-1}{\sqrt{2x-1}}$ into $2$ simpler fractions with common denominator $\sqrt{2x-1}$

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

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

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

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Learn how to solve integrals of rational functions problems step by step online. Find the integral int((x^2-1)/((2x-1)^1/2))dx. Expand the fraction \frac{x^2-1}{\sqrt{2x-1}} into 2 simpler fractions with common denominator \sqrt{2x-1}. Expand the integral \int\left(\frac{x^2}{\sqrt{2x-1}}+\frac{-1}{\sqrt{2x-1}}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{x^2}{\sqrt{2x-1}}dx results in: \frac{1}{20}\sqrt{\left(2x-1\right)^{5}}+\frac{1}{6}\sqrt{\left(2x-1\right)^{3}}+\frac{1}{4}\sqrt{2x-1}. Gather the results of all integrals.

Final Answer

$-\frac{3}{4}\sqrt{2x-1}+\frac{1}{6}\sqrt{\left(2x-1\right)^{3}}+\frac{1}{20}\sqrt{\left(2x-1\right)^{5}}+C_0$

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^2-1)/((2x-1)^0.5))dx using partial fractionsSolve integral of ((x^2-1)/((2x-1)^0.5))dx using basic integralsSolve integral of ((x^2-1)/((2x-1)^0.5))dx using u-substitutionSolve integral of ((x^2-1)/((2x-1)^0.5))dx using integration by partsSolve integral of ((x^2-1)/((2x-1)^0.5))dx using trigonometric substitution

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Function Plot

Plotting: $-\frac{3}{4}\sqrt{2x-1}+\frac{1}{6}\sqrt{\left(2x-1\right)^{3}}+\frac{1}{20}\sqrt{\left(2x-1\right)^{5}}+C_0$

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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
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: Integrals of Rational Functions

Integrals of rational functions of the form R(x) = P(x)/Q(x).

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