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This is a Objective answer type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}LetI=\int \frac{x^9}{{\left(4x^2+1\right)}^6}dx\\ ={\displaystyle \int \frac{x^9}{x^{12}{\left(4+\frac{1}{x^2}\right)}^6}}dx={\displaystyle \int \frac{1}{x^3{\left(4+\frac{1}{x^2}\right)}^6}}dx\\ Put\left(4+\frac{1}{x^2}\right)=t\Rightarrow \frac{-2}{x^3}dx=dt\Rightarrow \frac{dx}{x^3}=-\frac{1}{2}dt\\ \therefore I=-\frac{1}{2}{\displaystyle \int \frac{dt}{t^6}}=-\frac{1}{2}\times -\frac{1}{5}{t}^{-5}+C=\frac{1}{10}{\left(4+\frac{1}{x^2}\right)}^{-5}+C\\ Hence,correctoptionis\left(d\right).\end{array}$

School of Business Management (NIU) eligibility depends on the specialisation. For BBA Retail Operations, candidates must have passed Class 12, or Class 10 plus a 2-year Diploma or 3 years of vocational education. For Hospital & Health Management and Aviation, students must have a minimum of 50% marks in Class 12 from a recognised board.

4.13. Kindly go through the solution

Students looking forward to the BTech course at GIET Engineering College, which is now known as the Godavari Global University, should stay informed about the admission timeline. It helps in staying updated with the current news and avoid any delay. Dates of the accepted entrance exam such as AP-EAMCET should be checked regularly.

This is a Objective answer type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}LetI={\displaystyle \int e^x{\left(\frac{1-x}{1+x}\right)}^2 dx }\\ ={\displaystyle \int e^x\left[\frac{1+x^2-2x}{{\left(1+x^2\right)}^2}\right]}dx={\displaystyle \int e^x\left[\frac{1+x^2}{{\left(1+x^2\right)}^2}-\frac{2x}{{\left(1+x^2\right)}^2}\right]}dx\\ ={\displaystyle \int e^x\left[\frac{1}{1+x^2}-\frac{2x}{{\left(1+x^2\right)}^2}\right]}dx\\ Heref\left(x\right)=\frac{1}{1+x^2}\therefore f^{\text{'}}\left(x\right)=\frac{-2x}{{\left(1+x^2\right)}^2}\\ Using{\displaystyle \int e^x\left[f\left(x\right)+f^{\text{'}}\left(x\right)\right]}dx=e^x.f\left(x\right)+C\\ \therefore I=e^x.\frac{1}{1+x^2}+C=\frac{e^x}{1+x^2}+C\\ Hence,correctoptionis\left(a\right).\end{array}$

Yes, BTech course is available at Chouksey Engineering College. The institute offers BTech courses at the UG level. Candidate must meet the eligibility criteria to enrol for BTech course admission set by the college. Aspirants must pass Class 12 with a 45% aggregate. Chouksey Engineering College's admission process is entrance-based.

ITM Institute of Hotel Management, Navi Mumbai provides good placements to its graduating students annually in major hotel chains abroad. Check out the table below to know the key highlights of the ITM Institute of Hotel Management, Navi Mumbai placements in 2025:

This is a Objective answer type Questions as classified in NCERT Exemplar

The ITM Institute of Hotel Management, Navi Mumbai has had decent placement opportunities over the years. Check out the table below to know the key highlights of the ITM Institute of Hotel Management, Navi Mumbai placements in 2025:

This is a Objective answer type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}LetI={\displaystyle \int \frac{dx}{sin\left(x-a\right).sin\left(x-b\right)} }\\ Multiplyinganddividingbysin\left(b-a\right)\\ =\frac{1}{sin\left(b-a\right)}{\displaystyle \int \frac{sin\left(b-a\right)}{sin\left(x-a\right).sin\left(x-b\right)} dx}\\ =\frac{1}{sin\left(b-a\right)}{\displaystyle \int \frac{sin\left(x+b-x-a\right)}{sin\left(x-a\right).sin\left(x-b\right)} dx}\\ =\frac{1}{sin\left(b-a\right)}{\displaystyle \int \frac{sin\left[\left(x-a\right)-\left(x-b\right)\right]}{sin\left(x-a\right).sin\left(x-b\right)} dx}\\ =\frac{1}{sin\left(b-a\right)}{\displaystyle \int \frac{sin\left[\left(x-a\right)-\left(x-b\right)\right]}{sin\left(x-a\right).sin\left(x-b\right)} dx}\\ =\frac{1}{sin\left(b-a\right)}{\displaystyle \int \frac{sin\left(x-a\right)cos\left(x-b\right)-cos\left(x-a\right)sin\left(x-b\right)}{sin\left(x-a\right).sin\left(x-b\right)} dx}\\ =\frac{1}{sin\left(b-a\right)}{\displaystyle \int \frac{sin\left(x-a\right)cos\left(x-b\right)}{sin\left(x-a\right).sin\left(x-b\right)}-\frac{cos\left(x-a\right)sin\left(x-b\right)}{sin\left(x-a\right).sin\left(x-b\right)} dx}\\ =\frac{1}{sin\left(b-a\right)}{\displaystyle \int \left[\frac{cos\left(x-b\right)}{sin\left(x-b\right)}-\frac{cos\left(x-a\right)}{sin\left(x-a\right)}\right]}dx\\ =\frac{1}{sin\left(b-a\right)}{\displaystyle \int \left[cot\left(x-b\right)-cot\left(x-a\right)\right]}dx\\ =\frac{1}{sin\left(b-a\right)}\left[logsin\left(x-b\right)-logsin\left(x-a\right)\right]+C\\ =\frac{1}{sin\left(b-a\right)}.log\left|\frac{sin\left(x-b\right)}{sin\left(x-a\right)}\right|+C\\ I=cosec\left(b-a\right).log\left|\frac{sin\left(x-b\right)}{sin\left(x-a\right)}\right|+C\\ Hence,correctoptionis\left(c\right).\end{array}$

NIU offers multiple BBA specialisations to suit different career goals. These include Retail Operations, Hospital & Health Management, and Aviation. Each specialisation is designed with industry inputs, helping students develop focused skills and practical knowledge relevant to their chosen field, making them job-ready and competitive in their respective industries.

BBA programme at NIU spans three to four years, divided into six to eight semesters. The duration depends on the chosen specialisation and academic path. This flexible structure allows students to complete their degree while gaining in-depth knowledge and industry-relevant skills for careers in business, management, and related sectors.

Chouksey Engineering College offers BTech course at the UG level. The college offers five course specialisations. Students can check the list of specialisations below:

• BTech. in Computer Science Engineering
• BTech in AI & ML
• BTech in AI & DS
• BTech. in Electrical and Electronics Engineering
• BTech. in Mechanical Engineering
• BTech in ET & T

No, the correction facility is usually free of cost. However, if your correction leads to a change in category or test attempt (e.g., adding another test), additional payment may be required.

The ITM Institute of Hotel Management, Navi Mumbai is known for its excellent placement record for both domestic and international placements. Check out the tabulated data given below to know more about ITM Institute of Hotel Management, Navi Mumbai recent placements in 2025:

This is a Objective answer type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}LetI={\displaystyle \int \frac{cos2x-cos2\theta }{cosx-cos\theta } dx}\\ ={\displaystyle \int \frac{\left(2co{s}^{2}x-1\right)-\left(2co{s}^2\theta -1\right)}{cosx-cos\theta } dx}\\ ={\displaystyle \int \frac{2co{s}^{2}x-1-2co{s}^2\theta +1}{cosx-cos\theta } dx}\\ ={\displaystyle \int \frac{2co{s}^{2}x-2co{s}^2\theta }{cosx-cos\theta } dx}=2{\displaystyle \int \frac{co{s}^{2}x-co{s}^2\theta }{cosx-cos\theta } dx}\\ =2{\displaystyle \int \frac{\left(cosx+cos\theta \right)\left(cosx-cos\theta \right)}{\left(cosx-cos\theta \right)} dx}\\ =2{\displaystyle \int \left(cosx+cos\theta \right)dx=}2\left(sinx+cos\theta .x\right)+C\\ Hence,correctoptionis\left(a\right).\end{array}$

Admission to NYU Masters in Computer Science requires students to have an Undergraduate degree in Computer Science, Mathematics, or Engineering. Applicants with other UG degree are also considered. Additional CS requirements are:

• At least 1 year of university-level Science
• High-level working knowledge of a programming language such as C+
• A basic understanding of Computer fundamentals
• English language proficiency requirements
• GRE test score can be submitted, but there is a waiver for this, if students have achieved equivalent eligibility as mentioned by the university

Students seeking to get into the M.Tech programme offered at the university must satisfy the admission criteria for the course. M.Tech admission at DY Patil Institute of Technology is entrance-based. Students need to appear for the accepted entrance exams, i.e., GATE/MHTCET. The basic eligibility is to pass graduation with a valid aggregate from a recognised university. However, as per the general report, many universities accept candidates who have scores of 60-70% marks in their graduation.

Yes, Royal Global University BCom admissions are open. The university is currently accepting applications for BCom admissions for the academic year 2025-26. Candidates willing to take admission at the university for a BCom must apply online. The application process is conducted on the official website of the university. The application form must be submitted before the stipulated deadline.