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${\displaystyle \underset{0}{\overset{\pi }{\int }}\frac{x^{2}sinx\cdot cosx}{si{n}^{4}x+co{s}^{4}x}dx}$

$={\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{sinxcosx}{si{n}^{4}x+co{s}^{4}x}\left(x^2-{\left(\pi -x\right)}^2\right)dx}$

$={\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{sinx\cdot cosx\left(2\pi x-{\pi }^2\right)}{si{n}^{4}x+co{s}^{4}x}x}$

$=2\pi \cdot \frac{\pi }{4}{\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{sinxcosx}{si{n}^{4}x+co{s}^{4}x}dx}-{\pi }^2{\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{sinxcosx}{si{n}^{4}x+co{s}^{4}x}dx}$

$=-\frac{{\pi }^2}{2}{\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{sinxcosx}{si{n}^{4}x+co{s}^{4}x}dx}$

$=-\frac{{\pi }^2}{2}{\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\frac{1}{2}sin2x}{1-\frac{1}{2}si{n}^{2}2x}dx}$

$=-\frac{{\pi }^2}{2}{\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{sin2x}{2-si{n}^{2}2x}dx}$

$=-\frac{{\pi }^2}{2}{\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{sin2x}{1+co{s}^{2}2x}dx}$

Let cot2x = t

$=-\frac{{\pi }^2}{2}{\displaystyle \underset{1}{\overset{-1}{\int }}\frac{-\frac{1}{2}dt}{1+t^2}}$

$=-\frac{{\pi }^2}{4}{\displaystyle \underset{-1}{\overset{1}{\int }}\frac{dt}{1+t^2}}$

$=-\frac{{\pi }^2}{4}\cdot \frac{\pi }{2}=-\frac{{\pi }^2}{8}$

$\frac{120}{{\pi }^3}\left|-\frac{{\pi }^3}{8}\right|=15$

Yes, the University of Sunderland is a good option for Indian students planning to study in the UK. It is much more affordable compared to other universities. The living costs are low. Sunderland sits among the top 10 affordable cities in the UK. Student get a very similar environment to most universities. The university also maintains strong connections in the industry uses them to provide good career opportunities. 

Take $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}=\lambda$  

x = 2λ + 1, y = 3λ + 2, z = 4λ + 3

  $\overrightarrow{AB}$  = (α − 2)  $\widehat{i}$ + (β − 3) $\widehat{j}$ + (γ − 4) $\widehat{k}$  

Now,

(α − 2) ⋅ 2 + (β − 3) ⋅3 + (γ − 4) ⋅ 4 = 0

2α − 4 + 3β − 9 + 4γ −16 = 0

⇒ 2α + 3β + 4γ = 29

Some courses that provide high scope include BDes, BSc in Physics, BBA, MBBS, CAT, and BCA.

Which course is better after 10th completely depends on the personal career goals and preferences of students. Many students choose Science stream for better future opportunities. However, the trend of Diploma and vocational courses is also rising nowadays. 

  $f\left(x\right)=e^{x^3-3x+1}$

$f\text{'}\left(x\right)=e^{x^3-3x+1}$  (3x² − 3)

$e^{x^3-3x+1}$ = ⋅ 3(x −1)(x +1)

For x ∈ (−∞, −1], f '(x) ≥ 0

∴     f(x) is increasing function

∴     a = e^–∞ = 0 = f (−∞)

b = e⁻¹⁺³⁺¹ = e³ = f (−1)

∴     P(4, e³ + 2)

$d=\frac{\left(e^3+2\right)\left(e^{-3}\right)}{\sqrt[]{1+e^{-6}}}=\frac{1+2e^{-3}}{\sqrt[]{1+e^{-6}}}=\frac{1+2e^{-3}}{\sqrt[]{1+e^{-6}}}=\frac{e^3+2}{\sqrt[]{e^6+1}}$

Yes, you can fill out the CLAT Application Form in phases. There is an option to save in every step. However, it will be better if you fill the form in one go. For this, you should keep all the documents ready in hand. See if you fill the form in one go you don't have to worry about it. 

The questions asked in Logical reasoning are from Alphanumeric Series, Reasoning Analogies, Blood Relations, Coding Decoding, Odd One Out. Picture Series and Sequences, Pattern Series and Sequences, Seating Arrangements, Statement and Assumptions, Syllogism. Apply various 10h ständard mathematical operations on such information, including from areas such as ratios and proportions, basic algebra, Profit and Loss, Time & Work, Average, Speed and distance, Ratio and Proportion. 

$=8*7*6*5*4+\frac{9*8}{2}$

= 6756

The questions in the english section of RULET exam are asked from Tenses, Phrasal Verbs and idioms, etc. Aspirants can check the Shiksha portal for more details. The syllabus for this exam will be similar to other entrance exam. 

As per the UniRaj official website, RULET exam consists of 4 sections:

1. English Language,
2. Legal Reasoning,
3. Current Affairs & GK
4. Logical Aptitude & Quantitative Aptitude 

The detailed syllabus of RULET 2026 as provided by the University of Rajasthan through the official notification. Candidates looking to sit for the exam must practice the questions related to these sections to ace the exam in first attempt. 

(y – 2) = m (x – 8)

⇒   x-intercept

⇒     $\left(\frac{-2}{m}+8\right)$

⇒   y-intercept

⇒   (–8m + 2)

⇒   OA + OB = $\frac{-2}{m^2}$  + 8 – 8m + 2

$f\text{'}\left(m\right)=\frac{2}{m^2}-8=0$  

-> $m^2=\frac{1}{4}$

-> $m=\frac{-1}{2}$

-> $f\left(\frac{-1}{2}\right)=18$

->Minimum = 18

Take e^sinx = t (t > 0)

⇒ $t-\frac{2}{t}=2$

⇒   $\frac{t^2-2}{t}=2$

->t² – 2t – 2 = 0

->t² – 2t + 1 = 3

⇒   (t −1)² = 3

⇒   t = 1 ± $\sqrt[]{3}$  

⇒   t = 1 ± 1.73

⇒   t = 2.73 or –0.73 (rejected as t > 0)

⇒   e^{sin x} = 2.73

->log_e e^{sin x} = log_e 2.73

⇒ sin x = log_e 2.73 > 1

So no solution.

RULET entrance exam is conducted once in a year by the University of Rajasthan. This exam is conducted for admission to the undergraduate Law courses. Aspiring candidates will have to keep checking the official notification for the registration of the entrance exam. The exam is tentatively held in the month of May or June every year.

Candidates who do not belong to the Rajasthan state can also apply for the RULET exam. The candidates who wish to pursue the UG in Law courses from University of Rajasthan will have to sit for the RULET exam. The interested candidates will have to apply for the entrance exam before the last date. 

Candidates can check the list of documents required to fill the registration form for CLAT below

• Passport-Size Photograph 
• Scanned Signature 
• Class 10 Marksheet
• Class 12 Marksheet
• Category Certificate
• PwD Certificate
• Domicile Certificate
• Valid Photo ID Proof
• Email ID and Mobile Number

RULET is an entrance exam conducted by the University of Rajasthan. The RULET exam is for admission to the undergraduate Law courses. Interested candidates will be required to fill in the application form.