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No, the last date to apply for the Certificate programme at the National Institute of Computer Arts (NICA) in 2025 has not been announced yet. Students should regularly check the official NICA website to stay updated on the final application deadline and other important dates.

Yes, Shri Guru Ram Rai University offers a DPharma course. Students seeking admission to the course can check the below table to know the key highlights of this programme:

Note: The above information is taken from official sources. However, it is subject to change.

Yes, admission is currently open for the Certificate programme at the National Institute of Computer Arts (NICA) for 2025. Interested students who meet the basic eligibility, Class 12 pass and minimum age of 18 years, can apply directly through the official NICA website.

$LetI={\displaystyle {\int }_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}co{s}^2}xdx·-----(i)$

$\begin{array}{l}={\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}cos{}^2}\left(\frac{\pi }{2}-x\right)dx\{?{\displaystyle {\int }_0^{a}f}(x)={\displaystyle {\int }_0^{a}f}(a-x)dx.\\ I={\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}si{n}^{2}xdx}-----(ii)\\ Adding\left(i\right)\&\left(ii\right),\\ 2I={\displaystyle {\int }_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(co{s}^{2}x+si{n}^{2}x\right)}dx\\ ={\displaystyle {\int }_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}1}·dx\\ 2I={[x]}_0^{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=\frac{\pi }{2}-0\\ 2I=\frac{\pi }{2}\\ I=\frac{\pi }{}\end{array}$

$\begin{array}{l}Given,f\left(x\right)={\displaystyle {\int }_0^{x}t}sintdt\\ =t{\displaystyle {\int }_0^{x}sintdt}-{\displaystyle {\int }_0^x\frac{dt}{dt}}{\displaystyle \int sintdtdt}\\ ={\left[t\left(-cost\right)\right]}_0^x-{\displaystyle {\int }_0^x\left(-cost\right)}dt\\ =-\left[xcosx-0cos0\right]+{\left[sint\right]}_0^x\\ =-xcosx+\left[sinx-sin0\right]\\ =-xcosx+sinx\\ \therefore f\text{'}\left(x\right)=\frac{d}{dx}\left(-xcosx+sinx\right)\\ =-x\frac{d}{dx}cosx-cosx\frac{dx}{dx}+\frac{d}{dx}sinx\\ =-x\left(-sinx\right)-cosx+sinx\\ =xsinx\end{array}$

? Option (B) is correct.

$\begin{array}{l}Let,I={\displaystyle {\int }_{1/3}^1\frac{{\left(x-x^3\right)}^{1/3}}{x^4}}dx\\ ={\displaystyle {\int }_{1/3}^1{\left[\frac{x^3\left(\frac{1}{x^2}-1\right)}{x^4}\right]}^{1/3}}dx\\ ={\displaystyle {\int }_{1/3}^1\frac{x{\left(\frac{1}{x^2}-1\right)}^{1/3}}{x^4}}dx={\displaystyle {\int }_{1/3}^1\frac{{\left(\frac{1}{x^2}-1\right)}^{1/3}}{x^3}}dx\\ ={\displaystyle {\int }_{1/3}^1{\left(x^{-2}-1\right)}^{1/3}}{x}^{-3}dx\end{array}$

= 6

Option A is correct

$\begin{array}{l}Let={\displaystyle {\int }_1^2\left(\frac{1}{x}-\frac{1}{2x^2}\right)}{e}^{2x}dx\\ Let,2x=t\Rightarrow dx=\frac{dt}{2}\\ When,x=1,t=2\times 1=2\\ x=2,t=2\times 2=4\end{array}$

$\begin{array}{l}So,I={\displaystyle {\int }_2^4\left(\frac{1}{\left(t/2\right)}-\frac{1}{t\left(\frac{t}{2}\right)}\right)}{e}^t\frac{dt}{2}\\ ={\displaystyle {\int }_2^4\left(\frac{2}{t}-\frac{2}{t^2}\right)}{e}^t\frac{dt}{2}\end{array}$

$={\displaystyle {\int }_2^4\left(\frac{1}{t}+\frac{\left(-1\right)}{t^2}\right)}{e}^{t}dt$ is in the form

$\begin{array}{l}Let,={\displaystyle {\int }_{-1}^1\frac{dx}{x^2+2x+5.}}\\ ={\displaystyle {\int }_{-1}^1\frac{dx}{x^2+2x+1+4}}\\ ={\displaystyle {\int }_{-1}^1\frac{dx}{{\left(x+1\right)}^2+4}}={\displaystyle {\int }_{-1}^1\frac{dx}{{\left(x+1\right)}^2+2^2.}}\end{array}$

Let x + 1 = t $\Rightarrow$ dx = dt

When x = 1, t = 1 + 1 = 2

x = –1, t = –1 + 1 = 0

$\begin{array}{l}I\; ={\displaystyle {\int }_0^2\frac{dt}{t^2+2^2}}\\ =\frac{1}{2}{\left[ta{n}^{-1}\frac{t}{2}\right]}_0^2\\ =\frac{1}{2}\left[ta{n}^{-1}\frac{2}{2}-ta{n}^{-1}\frac{0}{2}\right]\\ =\frac{1}{2}\left(ta{n}^{-1}1-ta{n}^{-1}0\right)\end{array}$

$=\frac{1}{2}[\pi /4$
$\begin{array}{l}\frac{\mathrm{}}{}-0]\\ =\pi /8\end{array}$

The Tolani Maritime Institute (TMI) has association with various companies that offers placement to the students BSc students of the institute. The information about the highest and average placement offered to the students and a few more are given in the following table:

Kindly go through the solution

Kindly go through the solution

Kindly go through the solution

Certificate courses at the National Institute of Computer Arts (NICA) are designed with flexible durations ranging from as short as 1 day to as long as 24 months. Students can opt for either full-time or part-time study modes depending on their schedule and learning preferences, making NICA's offerings suitable for a wide range of learners.

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