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Find the integral $\int\frac{5x^2+6x-8}{\left(x-6\right)\left(x-2\right)^4\left(x+1\right)^3}dx$

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Solution

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Final Answer

$\frac{1}{422}\ln\left(x-6\right)+\frac{2}{27\left(x-2\right)^{3}}+\frac{-1}{126\left(x+1\right)^{2}}-\frac{19}{789}\ln\left(x-2\right)+\frac{-\frac{9}{434}}{x-2}+\frac{\frac{31}{754}}{\left(x-2\right)^{2}}+\frac{8}{373}\ln\left(x+1\right)+\frac{-\frac{13}{435}}{x+1}+C_0$
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Step-by-step Solution

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Rewrite the fraction $\frac{5x^2+6x-8}{\left(x-6\right)\left(x-2\right)^4\left(x+1\right)^3}$ in $8$ simpler fractions using partial fraction decomposition

$\frac{5x^2+6x-8}{\left(x-6\right)\left(x-2\right)^4\left(x+1\right)^3}=\frac{A}{x-6}+\frac{B}{\left(x-2\right)^4}+\frac{C}{\left(x+1\right)^3}+\frac{D}{x-2}+\frac{F}{\left(x-2\right)^{2}}+\frac{G}{\left(x-2\right)^{3}}+\frac{H}{x+1}+\frac{I}{\left(x+1\right)^{2}}$

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$\frac{5x^2+6x-8}{\left(x-6\right)\left(x-2\right)^4\left(x+1\right)^3}=\frac{A}{x-6}+\frac{B}{\left(x-2\right)^4}+\frac{C}{\left(x+1\right)^3}+\frac{D}{x-2}+\frac{F}{\left(x-2\right)^{2}}+\frac{G}{\left(x-2\right)^{3}}+\frac{H}{x+1}+\frac{I}{\left(x+1\right)^{2}}$

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Learn how to solve differential calculus problems step by step online. Find the integral int((5x^2+6x+-8)/((x-6)(x-2)^4(x+1)^3))dx. Rewrite the fraction \frac{5x^2+6x-8}{\left(x-6\right)\left(x-2\right)^4\left(x+1\right)^3} in 8 simpler fractions using partial fraction decomposition. Find the values for the unknown coefficients: A, B, C, D, F, G, H, I. The first step is to multiply both sides of the equation from the previous step by \left(x-6\right)\left(x-2\right)^4\left(x+1\right)^3. Multiplying polynomials. Simplifying.

Final Answer

$\frac{1}{422}\ln\left(x-6\right)+\frac{2}{27\left(x-2\right)^{3}}+\frac{-1}{126\left(x+1\right)^{2}}-\frac{19}{789}\ln\left(x-2\right)+\frac{-\frac{9}{434}}{x-2}+\frac{\frac{31}{754}}{\left(x-2\right)^{2}}+\frac{8}{373}\ln\left(x+1\right)+\frac{-\frac{13}{435}}{x+1}+C_0$

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 ((5x^2+6x)/(x-6)(x-2)^4)dx using partial fractionsSolve integral of ((5x^2+6x)/(x-6)(x-2)^4)dx using basic integralsSolve integral of ((5x^2+6x)/(x-6)(x-2)^4)dx using u-substitutionSolve integral of ((5x^2+6x)/(x-6)(x-2)^4)dx using integration by partsSolve integral of ((5x^2+6x)/(x-6)(x-2)^4)dx using trigonometric substitution

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Function Plot

Plotting: $\frac{1}{422}\ln\left(x-6\right)+\frac{2}{27\left(x-2\right)^{3}}+\frac{-1}{126\left(x+1\right)^{2}}-\frac{19}{789}\ln\left(x-2\right)+\frac{-\frac{9}{434}}{x-2}+\frac{\frac{31}{754}}{\left(x-2\right)^{2}}+\frac{8}{373}\ln\left(x+1\right)+\frac{-\frac{13}{435}}{x+1}+C_0$

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◻√◻
◻√
∞
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log
log◻
lim
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∫◻
|◻|
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: Differential Calculus

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.

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