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Find the integral $\int\frac{12x^5-35x^4+55x^3-78x^2+73x-33}{3x^2-5x+3}dx$

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Final Answer

$x^{4}-\frac{5}{3}x^{3}+3x^2-11x+C_0$
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Step-by-step Solution

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Divide $12x^5-35x^4+55x^3-78x^2+73x-33$ by $3x^2-5x+3$

$\begin{array}{l}\phantom{\phantom{;}3x^{2}-5x\phantom{;}+3;}{\phantom{;}4x^{3}-5x^{2}+6x\phantom{;}-11\phantom{;}\phantom{;}}\\\phantom{;}3x^{2}-5x\phantom{;}+3\overline{\smash{)}\phantom{;}12x^{5}-35x^{4}+55x^{3}-78x^{2}+73x\phantom{;}-33\phantom{;}\phantom{;}}\\\phantom{\phantom{;}3x^{2}-5x\phantom{;}+3;}\underline{-12x^{5}+20x^{4}-12x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-12x^{5}+20x^{4}-12x^{3};}-15x^{4}+43x^{3}-78x^{2}+73x\phantom{;}-33\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}-5x\phantom{;}+3-;x^n;}\underline{\phantom{;}15x^{4}-25x^{3}+15x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}15x^{4}-25x^{3}+15x^{2}-;x^n;}\phantom{;}18x^{3}-63x^{2}+73x\phantom{;}-33\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}-5x\phantom{;}+3-;x^n-;x^n;}\underline{-18x^{3}+30x^{2}-18x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-18x^{3}+30x^{2}-18x\phantom{;}-;x^n-;x^n;}-33x^{2}+55x\phantom{;}-33\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}-5x\phantom{;}+3-;x^n-;x^n-;x^n;}\underline{\phantom{;}33x^{2}-55x\phantom{;}+33\phantom{;}\phantom{;}}\\\phantom{;;;\phantom{;}33x^{2}-55x\phantom{;}+33\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\\\end{array}$

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$\begin{array}{l}\phantom{\phantom{;}3x^{2}-5x\phantom{;}+3;}{\phantom{;}4x^{3}-5x^{2}+6x\phantom{;}-11\phantom{;}\phantom{;}}\\\phantom{;}3x^{2}-5x\phantom{;}+3\overline{\smash{)}\phantom{;}12x^{5}-35x^{4}+55x^{3}-78x^{2}+73x\phantom{;}-33\phantom{;}\phantom{;}}\\\phantom{\phantom{;}3x^{2}-5x\phantom{;}+3;}\underline{-12x^{5}+20x^{4}-12x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-12x^{5}+20x^{4}-12x^{3};}-15x^{4}+43x^{3}-78x^{2}+73x\phantom{;}-33\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}-5x\phantom{;}+3-;x^n;}\underline{\phantom{;}15x^{4}-25x^{3}+15x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}15x^{4}-25x^{3}+15x^{2}-;x^n;}\phantom{;}18x^{3}-63x^{2}+73x\phantom{;}-33\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}-5x\phantom{;}+3-;x^n-;x^n;}\underline{-18x^{3}+30x^{2}-18x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-18x^{3}+30x^{2}-18x\phantom{;}-;x^n-;x^n;}-33x^{2}+55x\phantom{;}-33\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}-5x\phantom{;}+3-;x^n-;x^n-;x^n;}\underline{\phantom{;}33x^{2}-55x\phantom{;}+33\phantom{;}\phantom{;}}\\\phantom{;;;\phantom{;}33x^{2}-55x\phantom{;}+33\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\\\end{array}$

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Learn how to solve differential calculus problems step by step online. Find the integral int((12x^5-35x^455x^3-78x^273x+-33)/(3x^2-5x+3))dx. Divide 12x^5-35x^4+55x^3-78x^2+73x-33 by 3x^2-5x+3. Resulting polynomial. Expand the integral \int\left(4x^{3}-5x^{2}+6x-11\right)dx into 4 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int4x^{3}dx results in: x^{4}.

Final Answer

$x^{4}-\frac{5}{3}x^{3}+3x^2-11x+C_0$

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 ((12x^5+-35x^4)/(3x^2+-5x))dx using partial fractionsSolve integral of ((12x^5+-35x^4)/(3x^2+-5x))dx using basic integralsSolve integral of ((12x^5+-35x^4)/(3x^2+-5x))dx using u-substitutionSolve integral of ((12x^5+-35x^4)/(3x^2+-5x))dx using integration by partsSolve integral of ((12x^5+-35x^4)/(3x^2+-5x))dx using trigonometric substitution

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Plotting: $x^{4}-\frac{5}{3}x^{3}+3x^2-11x+C_0$

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x
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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: 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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