Chapter 15

The equation of a curve passing through origin is given by \(y=\int x^{3} \cos x^{4} d x .\) If the equation of the curve is written in the form \(x=g(y)\), then (A) \(g(y)=\sqrt[3]{\sin ^{-1}(4 y)}\) (B) \(g(y)=\sqrt{\sin ^{-1}(4 y)}\) (C) \(g(y)=\sqrt[4]{\sin ^{-1}(4 y)}\) (D) none of these

If \(\phi(x)=\int \frac{d x}{\sin ^{12} x \cos ^{1 / 2} x^{2}}\), then \(\phi\left(\frac{\pi}{4}\right)-\phi(0)=\) (A) \(\frac{12}{5}\) (B) \(\frac{9}{5}\) (C) \(\frac{6}{5}\) (D) 0

If \(\phi(x)=\lim _{n \rightarrow \infty} \frac{x^{n}-x^{-n}}{x^{n}+x^{-n}}, 0

\(\int \frac{\sin ^{8} x-\cos ^{8} x}{1-2 \sin ^{2} x \cos ^{2} x} d x\) is equal to \(\begin{array}{ll}\text { (A) } \frac{1}{2} \sin 2 x+c & \text { (B) }-\frac{1}{2} \sin 2 x+c\end{array}\) (C) \(-\frac{1}{2} \sin x+c\) (D) \(-\sin ^{2} x+c\)

If \(\int \frac{(\sqrt{x})^{5}}{(\sqrt{x})^{7}+x^{6}} d x=a \operatorname{In}\left(\frac{x^{k}}{x^{k}+1}\right)+c\), the values of \(a\) and \(k\) respectively are (A) \(\frac{5}{2}\) and \(\frac{2}{5}\) (B) \(\frac{2}{5}\) and \(\frac{5}{2}\) (C) \(\frac{5}{2}\) and 2 (D) none of there

\(\int x\left[f\left(x^{2}\right) g^{\prime \prime}\left(x^{2}\right)-f^{\prime \prime}\left(x^{2}\right) g\left(x^{2}\right)\right] d x\) (A) \(f\left(x^{2}\right) g^{\prime}\left(x^{2}\right)-g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)+c\) (B) \(\frac{1}{2}\left[f\left(x^{2}\right) g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)\right]+c\) (C) \(\frac{1}{2}\left[f\left(x^{2}\right) g^{\prime}\left(x^{2}\right)-g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)\right]+c\) (D) none of the above

The anti-derivative of \(\frac{\cos 5 x+\cos 4 x}{1-2 \cos 3 x}\) is (A) \(\frac{\sin 2 x}{2}+\cos x+c\) (B) \(-\frac{\sin 2 x}{2}+\sin x+c\) (C) \(-\frac{\sin 2 x}{2}-\sin x+c\) (D) \(\frac{\sin 2 x}{2}-\cos x+c\)

If \(\int \tan ^{4} x d x n=K \tan ^{3} x+L \tan x+f(x)\), then (A) \(K=\frac{1}{3}, L=-1, f(x)=x+C\) (B) \(K=1, L=-1, f(x)=-x+C\) (C) \(K=-1, L=1, f(x)=2 x+C\) (D) \(K=\frac{1}{2}, L=\frac{1}{3}, f(x)=3 x+C\)

\(\int \frac{1}{\left[(x-1)^{3}(x+2)^{5}\right]^{1 / 4}} d x\) is equal to (A) \(\frac{4}{3}\left(\frac{x-1}{x+2}\right)^{14}+c\) (B) \(\frac{4}{3}\left(\frac{x+2}{x-1}\right)^{14}+c\) (C) \(\frac{1}{3}\left(\frac{x-1}{x+2}\right)^{14}+c\) (D) \(\frac{1}{3}\left(\frac{x+2}{x-1}\right)^{14}+c\)

\(\int\left(\frac{\ln x-1}{(\ln x)^{2}+1}\right)^{2} d x\) is equal to (A) \(\frac{x}{x^{2}+1}+c\) (B) \(\frac{\ln x}{(\ln x)^{2}+1}+c\) (C) \(\frac{x}{(\ln x)^{2}+1}+c\) (D) \(e^{x}\left(\frac{x}{x^{2}+1}\right)+c\)

The value of \(\int \frac{a x^{2}-b}{x \sqrt{c^{2} x^{2}-\left(a x^{2}+b\right)^{2}}} d x\) is equal to (A) \(\sin ^{-1}\left(\frac{\left(a x+\frac{b}{x}\right)}{c}\right)+k\) (B) \(\sin ^{-1}\left(\frac{a x^{2}+\frac{b}{x^{2}}}{c}\right)+k\) (C) \(\cos ^{-1}\left(\frac{a x+\frac{b}{x}}{c}\right)+k\) (d) \(\cos ^{-1}\left(\frac{\left(a x^{2}+\frac{b}{x^{2}}\right)}{c}\right)+k\)

The value of \(\int \frac{\sec x d x}{\sqrt{\sin (2 x+\theta)+\sin \theta}}\) is (A) \(\sqrt{(\tan x+\tan \theta) \sec \theta}+c\) (B) \(\sqrt{2(\tan x+\tan \theta) \sec \theta}+c\) (C) \(\sqrt{2(\sin x+\tan \theta) \sec \theta}+c\) (D) none of these

\(\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2 x}} d x\) is equal to (A) \(\sin ^{-1}(\sin x+\cos x)+c\) (B) \(\sin ^{-1}\left[\frac{1}{3}(\sin x+\cos x)\right]+c\) (C) \(\cos ^{-1}(\sin x+\cos x)+c\) (D) none of these

If \(f(x)=\int \frac{x^{2} d x}{\left(1+x^{2}\right)\left(1+\sqrt{1+x^{2}}\right)}\) and \(f(0)=0\), then the value of \(f(1)\) is (A) \(\log (1+\sqrt{2})\) (B) \(\log (1+\sqrt{2})-\frac{\pi}{4}\) (C) \(\log (1+\sqrt{2})+\frac{\pi}{2}\) (D) none of these

If \(\int \frac{(x+1)}{x\left(1+x e^{x}\right)^{2}} d x=\log |1-f(x)|+f(x)+C\), then \(f(x)=\) (A) \(\frac{1}{x+e^{x}}\) (B) \(\frac{1}{1+x e^{x}}\) (C) \(\frac{1}{\left(1+x e^{x}\right)^{2}}\) (D) \(\frac{1}{\left(x+e^{x}\right)^{2}}\)

If \(\int f(x) \sin x \cos x d x=\frac{1}{2\left(b^{2}-a^{2}\right)} \log [f(x)]+C\), then \(f(x)\) is equal to (A) \(\frac{1}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x}\) (B) \(\frac{1}{a^{2} \sin ^{2} x-b^{2} \cos ^{2} x}\) (C) \(\frac{1}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}\) (D) \(\frac{1}{a^{2} \cos ^{2} x-b^{2} \sin ^{2} x}\)

\(\int \frac{d x}{(x+a)^{87}(x-b)^{67}}\) is equal to (A) \(\left(\frac{7}{a+b}\right)\left(\frac{x+a}{x-b}\right)^{17}+c\) (B) \(\left(\frac{7}{a+b}\right)\left(\frac{x-b}{x+a}\right)^{1 / 7}+c\) (C) \(\frac{6}{a+b}\left(\frac{x-b}{x+a}\right)^{16}+c\) (D) \(\frac{6}{a+b}\left(\frac{x+a}{x-b}\right)^{16}+c\)

\(\int \frac{\sqrt{x}}{\sqrt{x^{3}+4}} d x\) equals (A) \(\frac{2}{3} \ln \left(\frac{2}{\sqrt{x^{3}}-\sqrt{x^{3}-4}}\right)+C\) (B) \(\frac{2}{3} \ln \left(\frac{2}{\sqrt{x^{3}}-\sqrt{x^{3}-4}}\right)+C\) (C) \(\frac{2}{3} \ln \left(\frac{2}{\sqrt{x^{3}}-\sqrt{x^{3}-4}}\right)+C\) (D) none of these

\(\int \frac{[f(x) \cdot \phi(x)-\phi(x) \cdot \phi(x)]}{f(x) \cdot \phi(x)} \log \frac{f(x)}{\phi(x)} d x\) is equal to (A) \(\log \frac{\phi(x)}{f(x)}+k\) (B) \(\frac{1}{2}\left[\log \frac{\phi(x)}{f(x)}\right]^{2}+k\) (C) \(\frac{\phi(x)}{f(x)} \log \frac{\phi(x)}{f(x)}+k\) (D) none of these

If \(I=\int \frac{1}{2 p} \sqrt{\frac{p-1}{p+1}} d p=f(p)+c\), then \(f(p)\) is equal to (A) \(\frac{1}{2} \ln \left(p-\sqrt{p^{2}-1}\right)\) (B) \(\left(\frac{1}{2} \cos ^{-1} p+\frac{1}{2} \sec ^{-1} p\right)\) (C) \(\frac{1}{2} \ln \sqrt{p+\sqrt{p^{2}-1}}-\frac{1}{2} \sec ^{-1} p\) (D) none of these

If \(\int \frac{1}{x+x^{5}} d x=f(x)+c\), then \(\int \frac{x^{4}}{x+x^{5}} d x\) is equal to (A) \(\log |x|+f(x)+c\) (B) \(\log |x|-f(x)+c\) (C) \(x f(x)+c\) (D) none of these

\(\int \frac{\left(x^{2}-2\right) d x}{\left(x^{4}+5 x^{2}+4\right) \tan ^{-1}\left(\frac{x^{2}+2}{x}\right)}\) is (A) \(\log \left|\tan ^{-1} \sqrt{x+2}\right|+C\) (B) \(\log \left|\tan ^{-1}\left(x+\frac{2}{x}\right)\right|+C\) (C) \(\sin ^{-1}\left(\frac{x+2}{x}\right)+C\) (D) \(\tan ^{-1}\left(\frac{x+2}{x}\right)+C\)

The value of \(\int e^{x} \frac{1+n x^{n-1}-x^{2 n}}{\left(1-x^{n}\right) \sqrt{1-x^{2 n}}} d x\) is (A) \(e^{x} \frac{\sqrt{1-x^{n}}}{1-x^{n}}+C\) (B) \(e^{x} \frac{\sqrt{1+x^{2 \pi}}}{1-x^{2 n}}+C\) (C) \(e^{x} \frac{\sqrt{1-x^{2 n}}}{1-x^{2 n}}+C\) (D) \(e^{x} \frac{\sqrt{1-x^{2 n}}}{1-x^{n}}+C\)

\(\int \sqrt{\frac{\cos x-\cos ^{3} x}{1-\cos ^{3} x}} d x\) is equal to (A) \(\frac{2}{3} \sin ^{-1}\left(\cos ^{3 / 2} x\right)+c\) (B) \(\frac{2}{3} \sin ^{-1}\left(\cos ^{3 / 2} x\right)+\) (C) \(\frac{2}{3} \cos ^{-1}\left(\cos ^{3 / 2} x\right)+c\) (D) none of the above

\(\int \cos ^{-37} x \sin ^{-117} x d x=\) (A) \(\log \left|\sin ^{4 \pi} x\right|+c\) (B) \(\frac{4}{7} \tan ^{47} x+c\) (C) \(\frac{-7}{4} \tan ^{-47} x+c\) (D) \(\log \left|\cos ^{3 / 7} x\right|+c\) (e) \(\frac{7}{4} \tan ^{-47} x+c\)

If the integral \(\int \frac{5 \tan x}{\tan x-2} d x=x+a \ln |\sin x-2 \cos x|\) \(+k\), then \(a\) is equal to: (A) \(-1\) (B) \(-2\) (C) 1 (D) 2

The integral \(\int \frac{d x}{\left(a^{2}-b^{2} x^{2}\right)^{3 / 2}}\), equals: (A) \(\frac{x}{\sqrt{a^{2}-b^{2} x^{2}}}+C\) (B) \(\frac{x}{a^{2} \sqrt{a^{2}-b^{2} x^{2}}}+C\) (C) \(\frac{a x}{\sqrt{a^{2}-b^{2} x^{2}}}+C\) (D) \(\frac{x}{a^{2} \sqrt{a^{2}-b^{2} x^{2}}}+C\)

The value of \(\sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right)}\) is (A) \(x+\ln \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c\) (B) \(x-\ln \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c\) (C) \(x+\ln \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c\) (D) \(x-\ln \left|\cos \left(x-\frac{\pi}{4}\right)\right|+c\)

For a natural number \(n\), the value of the integral \(\int\left(x^{3 n}+x^{2 n}+x^{n}\right)\left(2 x^{2 n}+3 x^{n}+6\right)^{1 / n} d x\) is (A) \(\frac{1}{6 n}\left(2 x^{3 n}+3 x^{2 n}+6 x^{n}\right)^{1 / n}+C\) (B) \(\frac{1}{6 n}\left(2 x^{3 n}+3 x^{2 \pi}+6 x^{n}\right)^{1 / n+1}+C\) (C) \(\frac{1}{6(n+1)}\left(2 x^{3 n}+3 x^{2 n}+6 x^{n}\right)^{1 / n+1}+C\) (D) none of these

\(\int \frac{\sqrt{1+\sqrt[3]{x}}}{\sqrt[3]{x^{2}}} d x\) is equal to (A) \(\left(1+x^{1 / 3}\right)^{3 / 2}+C\) (B) \(-\left(1+x^{1 / 3}\right)^{3 / 2}+C\) (C) \(2\left(1+x^{1 / 3}\right)^{3 / 2}+C\) (D) none of these

\(\int x\left\\{f\left(x^{2}\right) g^{\prime \prime}\left(x^{2}\right)-f^{\prime \prime}\left(x^{2}\right) g\left(x^{2}\right)\right\\} d x=\) (A) \(f\left(x^{2}\right) g^{\prime}\left(x^{2}\right)-g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)+c\) (B) \(\frac{1}{2}\left\\{f\left(x^{2}\right) g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)\right\\}+c\) (C) \(\frac{1}{2}\left\\{f\left(x^{2}\right) g^{\prime}\left(x^{2}\right)-g\left(x^{2}\right) f^{\prime}\left(x^{2}\right)\right\\}+c\) (D) none of these

\(\frac{e^{x}\left(2-x^{2}\right)}{(1-x) \sqrt{1-x^{2}}} d x=\) (A) \(e^{x} \frac{\sqrt{1+x}}{\sqrt{1-x^{2}}}+C\) (B) \(e^{x} \frac{\sqrt{1-x}}{\sqrt{1+x}}+C\) (C) \(e^{x} \frac{\sqrt{1+x}}{\sqrt{1-x}}+C\) (D) none of these

\(\int\left\\{\left(\frac{x}{e}\right)^{x}+\left(\frac{e}{x}\right)^{x}\right\\} \ln x d x=\) (A) \(\left(\frac{x}{e}\right)^{x}-\left(\frac{e}{x}\right)^{x}+C\) (B) \(\left(\frac{x}{e}\right)^{x}+\left(\frac{e}{x}\right)^{x}+C\) (C) \(\left(\frac{x}{e}\right)^{x}-2\left(\frac{e}{x}\right)^{x}+C\) (D) none of these

\(\int\left(x^{3} a+x^{2} a+x a\right)\left(2 x^{2} a+3 x a+6\right)^{1} a d x=\) (A) \(\frac{1}{6(a+1)}\left(2 x^{3 a}+3 x^{2 a}+6 x^{a}\right)^{1-\frac{1}{a}}+C\) (B) \(\frac{1}{6(a+1)}\left(2 x^{3 a}+3 x^{2 a}+6 x^{a}\right)^{1+\frac{1}{a}}+C\) (C) \(\frac{1}{3(a+1)}\left(2 x^{3 a}+3 x^{2 a}+6 x^{a}\right)^{1+\frac{1}{a}}+C\) (D) none of these

\(\int \frac{\cos x}{1-\sin x \cos x} d x=\tan ^{-1}(\sin x-\cos x)\) \(+\frac{k}{\sqrt{3}} \ln \left|\frac{\sin x+\cos x-\sqrt{3}}{\sin x+\cos x-\sqrt{3}}\right|+C\), where \(k=\) (A) \(-\frac{1}{2}\) (B) \(\frac{1}{2}\) (C) \(-1\) (D) 1

\(\int \frac{\cos x}{1-\sin x \cos x} d x=\tan ^{-1}(\sin x-\cos x)\) \(+\frac{k}{\sqrt{3}} \ln \left|\frac{\sin x+\cos x-\sqrt{3}}{\sin x+\cos x-\sqrt{3}}\right|+C\), where \(k=\) (A) \(-\frac{1}{2}\) (B) \(\frac{1}{2}\) (C) \(-1\) (D) 1

\(\int \frac{\cos \left(x+\frac{\pi}{4}\right)}{2+\sin 2 x} d x\) (A) \(\sqrt{2} \tan ^{-1}(\sin x-\cos x)+C\) (B) \(\frac{1}{\sqrt{2}} \tan ^{-1}(\sin x-\cos x)+C\) (C) \(\frac{1}{\sqrt{2}} \tan ^{-1}(\sin x+\cos x)+C\) (D) \(\sqrt{2} \tan ^{-1}(\sin x+\cos x)+C\)

\(\int \frac{d x}{1-\cos ^{4} x}=-\frac{1}{2 \tan x}+\frac{k}{\sqrt{2}} \tan ^{-1}\left(\frac{\tan x}{\sqrt{2}}\right)+C\) where \(k=\) (A) \(\frac{1}{2}\) (B) \(-\frac{1}{2}\) (C) \(-1\) (D) 1

\(\int \frac{\sqrt{\cot x}-\sqrt{\tan x}}{1+3 \sin 2 x} d x\) \(=k \tan ^{-1}\left(\frac{\sqrt{\tan x}+\sqrt{\cot x}}{2}\right)+C\), where \(k=\) (A) 1 (B) \(-1\) (C) 2 (D) \(-2\)

\(\int \frac{\cos 5 x+\cos 4 x}{1-2 \cos 3 x} d x=k \sin x(1+\cos x)+C\), where \(k=\) (A) 2 (B) \(-2\) (C) 1 (D) \(-1\)

\(\int \frac{\sec x d x}{\sqrt{\sin (2 x+a)+\sin a}}=k \sqrt{\tan x+\tan a}+C\), where \(k=\) (A) \(\sqrt{\frac{2}{\cos a}}\) (B) \(\sqrt{2 \cos a}\) (C) \(\sqrt{\cos a}\) (D) \(\sqrt{\frac{1}{\cos a}}\)

\(\int \sqrt{x+\sqrt{x^{2}+2}} d x\) \(=\frac{1}{3}\left(\sqrt{x^{2}+2}+x\right)^{3 / 2}+k\left(\sqrt{x^{2}+2}-x\right)^{1 / 2}+C\), where \(k=\) (A) 2 (B) \(\sqrt{2}\) (C) \(-2\) (D) \(-\sqrt{2}\)

\(\int \frac{x^{4}-1}{x^{2} \sqrt{x^{4}+x^{2}+1}} d x=\) (A) \(\frac{\sqrt{x^{4}+x^{2}+1}}{x}+C\) (B) \(\frac{x}{\sqrt{x^{4}+x^{2}+1}}+C\) (C) \(-\frac{\sqrt{x^{4}+x^{2}+1}}{x}+C\) (D) none of these

\(\int \frac{x^{2}-1}{\left(x^{2}+1\right) \sqrt{1+x^{4}}} d x=k \cos ^{-1}\left(\frac{\sqrt{2} x}{x^{2}+1}\right)+C\), where \(k=\) (A) \(\frac{1}{2}\) (B) 2 (C) \(\frac{1}{\sqrt{2}}\) (D) \(\sqrt{2}\)

\(\int \frac{d x}{(1+\sqrt{x}) \sqrt{x-x^{2}}}=k\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}\right)+C\), where \(k=\) (A) \(]\) (B) 2 (C) 3 (D) 4

\(\int \frac{d x}{\cos ^{3} x \sqrt{\sin 2 x}}=\) (A) \(\sqrt{2}\left(\tan ^{1 / 2} x+\frac{1}{5} \tan ^{5 / 2} x\right)+C\) (B) \(\sqrt{2}\left(\cot ^{1 / 2} x+\frac{1}{5} \cot ^{5 / 2} x\right)+C\) (C) \(\sqrt{2}\left(\tan ^{1 / 2} x-\frac{1}{5} \tan ^{5 / 2} x\right)+C\) (D) none of these

\(\int \frac{1+x^{4}}{\left(1-x^{4}\right)^{3 / 2}} d x=\) (A) \(\frac{1}{\sqrt{x^{2}-\frac{1}{x^{2}}}}+c\) (B) \(\frac{1}{\sqrt{\frac{1}{x^{2}}-x^{2}}}+c\) (C) \(\frac{1}{\sqrt{x^{2}+\frac{1}{x^{2}}}}+c\) (D) none of these

\(\int \frac{\left(x^{2}-1\right)}{\left(x^{4}+3 x^{2}+1\right) \tan ^{-1}\left(x+\frac{1}{x}\right)} d x\) (A) \(\log \left|\tan ^{-1}\left(x+\frac{1}{x}\right)\right|+c\) (B) \(\log \left|\cot ^{-1}\left(x+\frac{1}{x}\right)\right|+c\) (C) \(2 \log \left|\tan ^{-1}\left(x+\frac{1}{x}\right)\right|+c\) (D) none of these

\(\int \sqrt{\frac{\cos x-\cos ^{3} x}{1-\cos ^{3} x}} d x=\) (A) \(\frac{2}{3} \sin ^{-1}\left(\cos ^{3 / 2} x\right)+c\) (B) \(-\frac{2}{3} \sin ^{-1}\left(\cos ^{3 / 2} x\right)+c\) (C) \(\frac{3}{2} \sin ^{-1}\left(\cos ^{3 / 2} x\right)+c\) (D) none of these

\(\int \frac{d x}{(x-1)^{3 / 4}(x+2)^{54}}=\) (A) \(\frac{4}{3}\left(\frac{x-1}{x+2}\right)^{14}+c\) (B) \(\frac{3}{4}\left(\frac{x-1}{x+2}\right)^{v 4}+c\) (C) \(\frac{4}{3}\left(\frac{x+2}{x-1}\right)^{14}+c\) (D) none of these