Problem 2
\(\int_{0}^{n / 2} \frac{\sqrt{\operatorname{ain} x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x\) in equal to (a) 0 (b) \(\underline{1}\) (c) \(\frac{\pi}{4}\) \((\) d \() \frac{5}{2}\).
Problem 4
\(\lim _{n \rightarrow-}\left[\frac{n}{n^{2}}+\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots+\frac{n}{n^{2}+(n-1)^{2}}\right]\) is equal to \((a)-\frac{\pi}{4}\) (b) 0 (c) \(\frac{\pi}{4}\) (d) \(\frac{\pi}{3}\)
Problem 5
\begin{aligned} &\int_{0}^{\pi / 2} \frac{\cos 2 x}{\cos x+\sin x} d x \text { equals }\\\ &\begin{array}{llll} (\text { a })-1 & \text { (b) } 0 & \text { (c) } 1 & \text { (d) } 2 \end{array} \end{aligned}
Problem 7
\(\int_{0}^{\pi} \sin ^{5}\left(\frac{x}{2}\right)\) is equal to (a) \(\frac{16}{15}\) (b) \(\frac{15}{16} \pi\) (c) \(\frac{16}{15} \pi^{2}\) (d) \(\frac{15}{16}\).
Problem 8
\(\int_{0}^{\pi / 2} \sin ^{99} x \cos x d x\) is equal to (a) \(\frac{1}{99}\) (b) \(\frac{\pi}{100}\) (c) \(\frac{99}{100}\) (d) None of these.
Problem 11
The area of the region in the firet quadrant bounded by the \(y\)-axis and the curves \(y=\sin x\) and \(y=\cos x\) is (a) \(\sqrt{2}\) (b) \(\sqrt{2}+1\) (c) \(\sqrt{2}-1\) (d) \(2 \sqrt{2}-1\)
Problem 12
The value of \(\int_{0}^{1} x^{3 / 2}(1-x)^{3 / 2} d x\) is (a) \(\pi=32\) (b) \(-\pi / 32\) \(\begin{array}{ll}\text { (c) } 3 \pi / 128 & (d)-3 \pi / 128\end{array}\)
Problem 15
If \(S_{1}\) and \(S_{2}\) are surface areas of the solids generated by revolving the ellipees \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) and \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) about the \(y\)-axis, then (a) \(S_{1}>S_{j}\) (b) \(S_{1}<\mathcal{S}_{2}\) (c) \(S_{1}=S_{2}\) (d) can't predict
Problem 18
\(\int_{0}^{2} x^{3} \sqrt{\left(2 x-x^{2}\right)} d x=\)
Problem 19
\(\int_{0}^{x / 2} \sin 2 \theta \log \tan \theta d \theta\) is equal te (a) 1 (b) \(-1\) (c) 0 (d) \(\pi / 2\),
