Integrate the function: \(\frac{5x}{(x+1)(x^2+9)}\)
Solution and Explanation
Let \(\frac{5x}{(x+1)(x^2+9)}\)\(=\frac{A}{(x+1)}+\frac{Bx+C}{(x^2+9)}...(1)\)
⇒5x=A(x2+9)+(Bx+C)(x+1)
⇒5x=Ax2+9A+Bx+Cx+C
Equating the coefficients of x2,x,and constant term,we obtain
A+B=0
B+C=5
9A+C=0
On solving these equations,we obtain
A=\(-\frac{1}{2}\),B=\(\frac{1}{2}\),and C=\(\frac{9}{2}\)
From equation(1),we obtain
\(\frac{5x}{(x+1)(x^2+9)}\)=\(-\frac{1}{2}\)(x+1)+\(\frac{\frac{x}{2}+\frac{9}{2}}{x^2+9}\)
=\(-\frac{1}{2}\)log|x+1|+\(\frac{1}{2}\)∫\(\frac{x}{x^2+9}\)dx+\(\frac{9}{2}\)∫\(\frac{1}{x^2+9}\)dx
=\(-\frac{1}{2}\)log|x+1|+\(\frac{1}{4}\)log(x2+9)+\(\frac{3}{2}\)tan-1\(\frac{x}{3}\)+C
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Questions Asked in CBSE CLASS XII exam
Read the following passage carefully:
(1) In recent years, air pollution has emerged as one of the most pressing environmental and public health issues in India. Cities like Delhi, Mumbai and Bengaluru often record Air Quality Indices (AQI) that fall into the ‘hazardous’ category, posing several health risks to their inhabitants. The thick blanket of smog that shrouds these urban areas is composed of harmful pollutants such as particulate material (PM 2.5 and PM 10), nitrogen dioxide (NO₂) and sulphur dioxide (SO₂). These pollutants originate from various sources including vehicular emissions, industrial activities and construction dust.
(2) The constant emission of pollutants from vehicles, industrial activities and construction works exacerbate the problem. Residents of these cities are often seen wearing masks and using air purifiers in a desperate attempt to mitigate the effects of polluted air. The ‘urban haze’ not only affects physical health but also impacts mental well-being. Studies indicate that prolonged exposure to high pollution levels can lead to respiratory issues, such as cardiovascular diseases and even cognitive decline.
(3) Furthermore, children and the elderly are the most vulnerable groups. Children, with their developing respiratory systems, and the elderly, often with pre-existing health conditions, face the brunt of this environmental crisis. Schools frequently shut down during peak pollution periods, disrupting education and daily life. The economic cost of air pollution is staggering, with increased healthcare expenses and lost labour productivity. Businesses also suffer as employees’ health issues lead to increased absenteeism and decreased efficiency.
(4) Government initiatives, such as the National Clean Air Programme (NCAP), aim at reducing air pollution levels across the country. However, these measures require robust implementation and public cooperation to be effective. Simple actions like reducing vehicle use, promoting public transportation and adopting cleaner technologies can significantly alleviate the situation. Additionally, the enforcement of stricter emission norms for industries and vehicles is crucial. Public awareness campaigns can play a pivotal role in encouraging others to adopt environmentally friendly practices.
(5) The role of technology in combating air pollution is also noteworthy. Innovative solutions such as air quality monitoring systems, green buildings and electric vehicles are gaining attraction. These technologies not only help in reducing pollution but also promote sustainable development. Urban planning that incorporates green spaces can also mitigate the effects of pollution by improving air quality and providing residents with a healthier living environment.
(6) As we navigate this challenge, it is imperative to remember that the solution lies not just in policy changes but also in the behavioural shifts of individuals. Embracing a lifestyle that prioritizes environmental health will pave the way for a cleaner, healthier India. Collective efforts and sustained commitments are crucial for ensuring breathable air for future generations. It is a shared responsibility that requires action from every segment of society, from policymakers to ordinary citizens.
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Answer the following questions, based on the above passage:
- CBSE CLASS XII - 2025
- Reading Comprehension
Read the following passage carefully :
(1) Introduction :In the digital age, social media has become an integral part of daily life, profoundly affecting how individuals interact and perceive the world. This case study aims to analyze the impact of social media on the mental health of different age groups in Indian society and explore the implications for overall well-being.
(2) Methodology : A survey was conducted among individuals across various age groups, ranging from teenagers to seniors, to determine their social media usage patterns. Participants were asked to indicate
the amount of time they spend on social media platforms and describe their feelings and experiences related to its use. The data was then analyzed to identify trends and patterns among different age demographics.(3) Survey Examination : The survey encompassed a comprehensive examination of social media habits, including not only the frequency and duration of usage but also the emotional and psychological effects of social media engagement. Participants were encouraged to articulate their experiences, providing invaluable insights into the multifaceted nature of social media's impact on mental health.
(4) Results : The survey results revealed interesting insights into social media usage and its effects on mental health among different age groups. Teenagers are deeply entrenched in exploring their self image and forming their identity. They are heavily reliant on their peer relationships and are vulnerable to the social aspects of online platforms to connect with friends and share experiences. As a result, their usage is higher that of other age groups.
(5) Implications for Interventions : By discerning the diverse effects of social media among different age groups, policy makers and mental health practitioners can tailor interventions to address specific demographic needs, e.g., educational campaigns aimed at teenagers could empower them to navigate social media more responsibly, while support groups for seniors can enhance their digital literacy and
reduce feelings of loneliness.(6) Conclusion : The survey outcomes serve as a roadmap for designing targeted interventions that not only cater to diverse demographic needs but also promote mental well-being. By harnessing the insights
gleaned from this study, stakeholders can collaboratively work towards building healthier online communities and promoting sustainable digital practices for generations to come.Answer the following questions, based on the above passage :
- CBSE CLASS XII - 2025
- Reading Comprehension
Read the following extracts and answer the questions for any one of the given two, (a) or (b) :
Those who prepare green wars,
wars with gas, wars with fire,
victory with no survivors,
would put on clean clothes
and walk about with their
brothers
in the shade, doing nothing.(Keeping Quiet)
A thing of beauty is a joy forever
Its loveliness increases, it will never
Pass into nothingness; but will keep
A bower quiet for us, and a sleep
Full of sweet dreams, and health, and quiet breathing.(A Thing of Beauty)
My Antarctic experience was full of such epiphanies, but the best occurred just short of the Antarctic Circle at 65.55 degrees south. The Shokalskiy had managed to wedge herself into a thick white stretch of ice between the peninsula and Tadpole Island which was preventing us from going any further. The Captain decided we were going to turn around and head back north, but before we did, we were all instructed to climb down the gangplank and walk on the ocean. So there we were, all 52 of us, kitted out in Gore-Tex and glares, walking on a stark whiteness that seemed to spread out forever. Underneath our feet was a metre-thick ice pack, .... In the periphery Crabeater seals were stretching and sunning themselves ... much like stray dogs .... It was nothing short of a revelation.
(Journey to the end of the Earth)
- CBSE CLASS XII - 2025
- Journey to the end of the earth
Concepts Used:
Definite Integral
Definite integral is an operation on functions which approximates the sum of the values (of the function) weighted by the length (or measure) of the intervals for which the function takes that value.
Definite integrals - Important Formulae Handbook
A real valued function being evaluated (integrated) over the closed interval [a, b] is written as :
\(\int_{a}^{b}f(x)dx\)
Definite integrals have a lot of applications. Its main application is that it is used to find out the area under the curve of a function, as shown below:
