Mathematics

# Integrate:$\displaystyle\int{\dfrac{{{\left( x-1 \right)}^{2}}}{{{x}^{3}}-3{{x}^{2}}+3x+8}}dx$

##### SOLUTION

Consider the given integral.

$I=\displaystyle\int{\dfrac{{{\left( x-1 \right)}^{2}}}{{{x}^{3}}-3{{x}^{2}}+3x+8}}dx$

Let $t={{x}^{3}}-3{{x}^{2}}+3x+8$

$\dfrac{dt}{dx}=3{{x}^{2}}-6x+3$

$\dfrac{dt}{3}={{x}^{2}}-2x+1$

$\dfrac{dt}{3}={{\left( x-1 \right)}^{2}}$

Therefore,

$I=\dfrac{1}{3}\displaystyle\int{\dfrac{1}{t}}dt$

$I=\dfrac{1}{3}\ln \left( t \right)+C$

On putting the value of $t$, we get

$I=\dfrac{1}{3}\ln \left( {{x}^{3}}-3{{x}^{2}}+3x+8 \right)+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Hard
A function $f$ is defined by $\displaystyle f(x)=\frac{1}{2^{r-1}},\frac{1}{2r}<x\leq \frac{1}{2^{r-1}},r=1,2,3,.....$ then the value of $\displaystyle \int _{0}^{1}f(x)dx$ is equal
• A. $\cfrac 13$
• B. $\cfrac 14$
• C. $\cfrac 12$
• D. $\cfrac 23$

1 Verified Answer | Published on 17th 09, 2020

Q2 Assertion & Reason Hard
##### ASSERTION

$\displaystyle \int_{0}^{1/8}\frac{dx}{1+\left ( \cot 4\pi x \right )^{\sqrt{2}}}=\frac{1}{8}$

##### REASON

If $\displaystyle a+b=\frac{\pi }{2}$, then $\displaystyle \int_{a}^{b}\frac{f\left ( \sin x \right )dx}{f\left ( \sin x \right )+f\left ( \cos x \right )}=\frac{b-a}{2}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Prove that $\displaystyle\int^{\pi}_0\sin^2x\cos^3xdx=0$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Integration of
$\displaystyle \int{\dfrac{x^{2}}{1+y^{2}}}dx$

Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$