\(\int_{0}^{1}\frac{(\tan ^{-1}x)^2}{1+x^2}\) dx =
  • 1
  • \(\frac{π^2}{64}\)
  • \(\frac{π^2}{192}\)
  • None of these
∫\(\frac{\sin ^2x-\cos ^2x}{\sin ^2xcos^2x}\) dx is equal to
  • tan x + cos x + c
  • tan x + cosec x + c
  • tan x + cot x + c
  • tan x+ sec x + c
What is the value of \(\int_{0}^{\pi / 2}\) \(\frac{\sqrt{\tan x}}{\sqrt{\tan x}+\sqrt{\cot x}}\) dx?
  • \(\frac{π}{2}\)
  • \(\frac{π}{4}\)
  • \(\frac{π}{8}\)
  • None of these
What is the value of \(\int_{0}^{\pi / 2}\) \(\frac{\sin x – \cos x}{1+\sin xcos x}\) dx?
  • 1
  • \(\frac{π}{2}\)
  • 0
  • –\(\frac{π}{2}\)
What is the value of \(\int_{\pi / 6}^{\pi / 3}\) \(\frac{dx}{\sin 2x}\)?
  • \(\frac{1}{2}\) log(-l)
  • log(- 1)
  • log 3
  • log √3
∫\(\frac{\sin x + \cos x}{\sqrt{1+2\sin x}}\) dx =
  • log(sin x – cos x)
  • x
  • log x
  • log sin (cos x)
∫(\(\frac{\cos 2θ – 1}{\cos 2θ + 1}\)) dθ =
  • tan θ – θ + c
  • θ + tan θ + c
  • θ – tan θ + c
  • -θ – cot θ + c
∫\(\frac{\cos 2x- \cos 2θ}{\cos x – \cos θ}\)dx is equal to
  • 2 (sin x + x cos θ) + C
  • 2 (sin x – x cos θ) + C
  • 2 (sin x + 2x cos θ) + C
  • 2 (sin x – 2x cos θ) + C
\(\int_{-\pi / 4}^{\pi / 4}\) \(\frac{dx}{1+\cos 2x}\) dx is equal to
  • 1
  • 2
  • 3
  • 4
\(\int_{0}^{\pi / 2}\) \(\sqrt{1+\sin 2x}\) dx is equal to
  • 2√2
  • 2(√2 + 1)
  • 0
  • 2(√2 – 1)
∫ \(\frac{a}{(1+x^2)\tan ^{-1}x}\) dx =
  • a log |tan x| + C
  • \(\frac{1}{2}\)(tanx)² + C
  • a log (1 + x) + C
  • None of these
∫ \(\frac{dx}{1-\cos x-\sin x}\) is equal to
  • log |1 + cot\(\frac{x}{2}\)| + C
  • log |1 – tan\(\frac{x}{2}\)| + C
  • log |1 – cot\(\frac{x}{2}\)| + C
  • log |1 + tan\(\frac{x}{2}\)| + C
Evaluate: ∫ \(\frac{1-\cos x}{\cos x(1+\cos x)}\) dx
  • log|sec x + tan x| – 2 tan(x/2) + C
  • log|sec x – tan x| – 2 tan(x/2) + C
  • log|sec x + tan x| + 2 tan(x/2) + C
  • None of these
∫\(\frac{dx}{\sin (x-a)\sin (x-b)}\) is equal to
  • sin(b – a) log |\(\frac{\sin (x-b)}{\sin (x-a)}\)| + C
  • cosec (b – a) log |\(\frac{\sin (x-b)}{\sin (x-b)}\)| + C
  • cosec (b – a) log |\(\frac{\sin (x-b)}{\sin (x-a)}\)| + C
  • sin (b – a) log |\(\frac{\sin (x-a)}{\sin (x-b)}\)| + C
∫ \(\frac{\cot x}{\sqrt[3]{\sin x}}\) dx =
  • None of these
  • \(\frac{-3}{\sqrt[3]{\sin x}}\) + C
  • \(\frac{-2}{\sin ^3 x}\) + C
  • \(\frac{3}{\sin ^{1/3}x}\) + C
Evaluate: ∫ \(\frac{1}{1+3\sin ^2x+8\cos ^2x}\) dx
  • \(\frac{1}{6}\) tan(2 tan x) + C
  • tan(2 tan x) + C
  • None of these
  • \(\frac{1}{6}\) tan(\(\frac{2 \tan x}{3}\)) + C
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