Mathematics

# Evaluate:$\displaystyle\int{\dfrac{{x}^{2}}{1+{x}^{3}}}dx$

##### SOLUTION
$\begin{array}{l}\text { Let } 1+x^{3}=z \\\quad \Rightarrow 3 x^{2} d x=d z \quad \Rightarrow x^{2} \mathrm{d} x=\dfrac{d z}{3} \\\therefore \quad \displaystyle \int \dfrac{x^{2}}{1+x^{3}} \mathrm{d} \mathbf{x}\\=\dfrac{1}{3} \displaystyle \int \dfrac{\mathrm{d} z}{z}\\=\dfrac{1}{3} \log |z|+C\\=\dfrac{1}{3} \log \left|1+x^{3}\right|+C\end{array}$

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

#### Realted Questions

Q1 Subjective Medium
Evaluate the integrals
$\int _{ 0 }^{ 2 }{ \cfrac { 1 }{ \left( { x }^{ 2 }+1 \right) } dx }$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Let $\displaystyle f(x)=\int{\frac{x^2}{(1+x^2)(1+\sqrt{1+x^2})}dx}$ and $f(0)=0$. Then $f(1)$ is equal to
• A. $\log_e{(1+\sqrt{2})}$
• B. $\displaystyle\log_e{(1+\sqrt{2})}+\frac{\pi}{4}$
• C. none of these
• D. $\displaystyle\log_e{(1+\sqrt{2})}-\frac{\pi}{4}$

1 Verified Answer | Published on 17th 09, 2020

Q3 One Word Medium
Evaluate:$\displaystyle\int \frac{dx}{\sqrt{4x^{2}-1}}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle\int_{0}^{\pi}\frac{dx}{1+a^{\cos x}}$ equals

• A. $0$
• B. $\displaystyle \pi$
• C. None of these
• D. $\displaystyle\frac{\pi }{2}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\int { (\cfrac { { 2 }^{ x }-5^{ x } }{ 10^{ x } } )dx }$ is equal to __________________.
• A. $\cfrac { { 2 }^{ x } }{ \log _{ e }{ 2 } } -\cfrac { 5^{ x } }{ \log _{ e }{ 5 } } +c$
• B. $\cfrac { { 2 }^{ x } }{ \log _{ e }{ 2 } } +\cfrac { 5^{ x } }{ \log _{ e }{ 5 } } +c$
• C. $\cfrac { { 5 }^{ -x } }{ \log _{ e }{ 5 } } -\cfrac { 2^{ -x } }{ \log _{ e }{ 2 } } +c$
• D. $\cfrac { { 2 }^{ -x } }{ \log _{ e }{ 2 } } -\cfrac { 5^{ -x } }{ \log _{ e }{ 5 } } +c$
The content structure of the given text passage is a mathematical question in the form of an integral expression, followed by a detailed solution to the problem.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Mathematics

# Evaluate:$\displaystyle\int{\dfrac{{x}^{2}}{1+{x}^{3}}}dx$

##### SOLUTION
$\begin{array}{l}\text { Let } 1+x^{3}=z \\\quad \Rightarrow 3 x^{2} d x=d z \quad \Rightarrow x^{2} \mathrm{d} x=\dfrac{d z}{3} \\\therefore \quad \displaystyle \int \dfrac{x^{2}}{1+x^{3}} \mathrm{d} \mathbf{x}\\=\dfrac{1}{3} \displaystyle \int \dfrac{\mathrm{d} z}{z}\\=\dfrac{1}{3} \log |z|+C\\=\dfrac{1}{3} \log \left|1+x^{3}\right|+C\end{array}$

Its FREE, you're just one step away

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

#### Realted Questions

Q1 Subjective Medium
Evaluate the integrals
$\int _{ 0 }^{ 2 }{ \cfrac { 1 }{ \left( { x }^{ 2 }+1 \right) } dx }$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Let $\displaystyle f(x)=\int{\frac{x^2}{(1+x^2)(1+\sqrt{1+x^2})}dx}$ and $f(0)=0$. Then $f(1)$ is equal to
• A. $\log_e{(1+\sqrt{2})}$
• B. $\displaystyle\log_e{(1+\sqrt{2})}+\frac{\pi}{4}$
• C. none of these
• D. $\displaystyle\log_e{(1+\sqrt{2})}-\frac{\pi}{4}$

1 Verified Answer | Published on 17th 09, 2020

Q3 One Word Medium
Evaluate:$\displaystyle\int \frac{dx}{\sqrt{4x^{2}-1}}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle\int_{0}^{\pi}\frac{dx}{1+a^{\cos x}}$ equals

• A. $0$
• B. $\displaystyle \pi$
• C. None of these
• D. $\displaystyle\frac{\pi }{2}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\int { (\cfrac { { 2 }^{ x }-5^{ x } }{ 10^{ x } } )dx }$ is equal to __________________.
• A. $\cfrac { { 2 }^{ x } }{ \log _{ e }{ 2 } } -\cfrac { 5^{ x } }{ \log _{ e }{ 5 } } +c$
• B. $\cfrac { { 2 }^{ x } }{ \log _{ e }{ 2 } } +\cfrac { 5^{ x } }{ \log _{ e }{ 5 } } +c$
• C. $\cfrac { { 5 }^{ -x } }{ \log _{ e }{ 5 } } -\cfrac { 2^{ -x } }{ \log _{ e }{ 2 } } +c$
• D. $\cfrac { { 2 }^{ -x } }{ \log _{ e }{ 2 } } -\cfrac { 5^{ -x } }{ \log _{ e }{ 5 } } +c$
user
The content structure of the given text passage is a mathematical question in the form of an integral expression, followed by a detailed solution to the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER