JEE Maths - Indefinite Integration

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Question - 1

If \(\int x \sqrt{\frac{2 \sin \left(x^{2}+1\right)-\sin 2\left(x^{2}+1\right)}{2 \sin \left(x^{2}+1\right)+\sin 2\left(x^{2}+1\right)}} d x =\log f\left(m\left(x^{2}+1\right)\right)+c \)  then evaluate \(\left[f\left(\frac{m \pi}{2}\right)\right]\) where [] represents the greatest integer function.

  • A 0
  • B 2
  • C 1
  • D 3

Question - 2

If \(\int(4 x+1) \sqrt{x^{2}-x-2} d x =\frac{4}{3} f\left(x^{2}-x-2\right)+\frac{p}{q}(2 x-1) g\left(x^{2}-x-2\right) -\left(\frac{m}{n}\right) h\left(\left|x-\frac{1}{2}+\sqrt{x^{2}-x-2}\right|\right)+c, \) then evaluate \(\left[\frac{m}{n}\right]+f(4)+p q+g(4)+h(1)\) where [ ] represents the greatest integer function, and G.C.D. (p, q) = 1, G.C.D.(m, n) = l.

  • A 10
  • B 25
  • C 20
  • D 35

Question - 3

If \(\int \frac{1}{\left(1+x^{2}\right) \sqrt{1-x^{2}}} d x=p f\left(\frac{\sqrt{1-x^{2}}}{q x}\right)+c\) then find the value of pq + tan (f (l)).

  • A 0
  • B 1
  • C 2
  • D 3

Question - 4

\(\text { Let } f(x)=\underbrace{x}_{\left(1+x^{n}\right)^{\frac{1}{n}}}, n \geq 2, g(x)=(\underbrace{f \circ f \circ_{\ldots 0} f}_{n \text { times }}(x) \)\(\text { and I }(n, x)=\int g(x) \cdot x^{n-2} d x \text { If I }(10,1)=\frac{(p)^{\frac{m}{k}}}{q}\) then find the value of  l0k - m - q + p. (Given that G.C.D. (m, k)= l)

  • A 11
  • B 12
  • C 24
  • D 32

Question - 5

If \(\int \sin ^{2} x \cos ^{4} x d x =\frac{p}{32}\left[\frac{\sin 6 x}{m}+\frac{\sin 4 x}{n}+\frac{\sin 2 x}{k}+q x\right]+c, \) then evaluate pm + knq

  • A 2
  • B 1
  • C 4
  • D 5

Question - 6

The graph of y = P (x), where P (x) is a polynomial, passes through (0, l) and increases in (2,3) and (4, -) and the decreases in (- \(\infty\),2) and (3, 4). If P (2) = - 15 and range of values of P(x) is fa, \(\infty\)), then evaluate P (3) - a.

  • A 0.25
  • B 0.5
  • C 0.75
  • D 0.55

Question - 7

If \(\int \frac{\sin x}{\sin (x-\alpha)} d x=A x+B \log \sin (x-\alpha)+C\) then value of (A, B) is ______________

  • A (- cos \(\alpha\), sin \(\alpha\))
  • B (cos \(\alpha\), sin \(\alpha\))
  • C (-sin \(\alpha\), cos \(\alpha\))
  • D (sin \(\alpha\),cos \(\alpha\))

Question - 8

If \(f(x)=\lim _{n \rightarrow \infty}\left[2 x+4 x^{3}+\ldots \ldots \ldots+2 n x^{2 n-1}\right]\) \((0 is equal to

  • A \(-\sqrt{1-x^{2}}+c\)
  • B \(\frac{1}{\sqrt{1-x^{2}}}+c\)
  • C \(\frac{1}{x^{2}-1}+c\)
  • D \(\frac{1}{1-x^{2}}+c\)

Question - 9

\( \int\left(\frac{f(x) g^{\prime}(x)-f^{\prime}(x) g(x)}{f(x) g(x)}\right)(\log (g(x)) -\log (f(x))) d x \) is equal to

  • A \(\log \left(\frac{g(x)}{f(x)}\right)+C\)
  • B \(\frac{1}{2}\left(\frac{g(x)}{f(x)}\right)^{2}+C\)
  • C \(\frac{1}{2}\left(\log \left(\frac{g(x)}{f(x)}\right)\right)^{2}+C\)
  • D \(\log \left(\left(\frac{g(x)}{f(x)}\right)^{2}\right)+C\)

Question - 10

\(\int e^{x} \frac{1+n x^{n-1}-x^{2 n}}{\left(1-x^{n}\right) \sqrt{1-x^{2 n}}} d x=\)

  • A \(e^{x}\left(\frac{1-x^{n}}{1+x^{n}}\right)+C\)
  • B \(e^{x}\left(\frac{1+x^{n}}{1-x^{n}}\right)+C\)
  • C \(e^{x}\left(\sqrt{\frac{1+x^{n}}{1-x^{n}}}\right)+C\)
  • D \(e^{x}\left(\sqrt{\frac{1-x^{n}}{1+x^{n}}}\right)+C\)
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