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Find the integral $\int\frac{x^2}{\left(x-1\right)\left(x-3\right)\left(x-5\right)}dx$

Step-by-step Solution

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Solution

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Final Answer

$\frac{1}{8}\ln\left(x-1\right)-\frac{9}{4}\ln\left(x-3\right)+\frac{25}{8}\ln\left(x-5\right)+C_0$
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Step-by-step Solution

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

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

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

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Learn how to solve equations with square roots problems step by step online. Find the integral int((x^2)/((x-1)(x-3)(x-5)))dx. Rewrite the fraction \frac{x^2}{\left(x-1\right)\left(x-3\right)\left(x-5\right)} in 3 simpler fractions using partial fraction decomposition. 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-1\right)\left(x-3\right)\left(x-5\right). Multiplying polynomials. Simplifying.

Final Answer

$\frac{1}{8}\ln\left(x-1\right)-\frac{9}{4}\ln\left(x-3\right)+\frac{25}{8}\ln\left(x-5\right)+C_0$

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-1)(x-3))dx using partial fractionsSolve integral of ((x^2)/(x-1)(x-3))dx using basic integralsSolve integral of ((x^2)/(x-1)(x-3))dx using u-substitutionSolve integral of ((x^2)/(x-1)(x-3))dx using integration by partsSolve integral of ((x^2)/(x-1)(x-3))dx using trigonometric substitution

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Function Plot

Plotting: $\frac{1}{8}\ln\left(x-1\right)-\frac{9}{4}\ln\left(x-3\right)+\frac{25}{8}\ln\left(x-5\right)+C_0$

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(◻)
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◻/◻
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÷
◻2
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√◻
√

◻√◻
◻√
∞
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: Equations with Square Roots

Those equations that they degree (greatest exponent) equals 2.

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