Mathematics - Indefinite Integration

Question - 1

If \(\int f(x)dx=f(x)\), then \(\int \left\{ {f(x)}^2dx \right\}\) is equal to

  • A \({1\over 2}\int \left\{ {f(x)} \right\}^2\)
  • B \({f \left\{(x)\right\}}^3\)
  • C \({f \left\{(x)\right\}}^3\over 3\)
  • D \({f \left\{(x)\right\}}^2\)

Question - 2

If \(\int{{1-x^7}\over x(1+x^7)}dx=\alpha\ ln\ |x|+b\ ln\ |x^7+1|+C,\) then

  • A \(a=1, b={2\over 7}\)
  • B \(a=-1, b={2\over 7}\)
  • C \(a=1, b=-{2\over 7}\)
  • D \(a=-1, b=-{2\over 7}\)

Question - 3

 If \(\int e^x \left\{ {f(x)-f'(x)} \right\} dx=\phi(x),\) then \(\int e^x{f(x)}dx\) is equal to

  • A \(\phi(x)=e^xf(x)\)
  • B \(\phi(x)-e^xf(x)\)
  • C \({1\over 2}\left\{\phi(x)+e^xf(x)\right\}\)
  • D \({1\over 2}\left\{\phi(x)+e^xf'(x)\right\}\)

Question - 4

If \({d\over dx}[f(x)]=xcos\ x+sin\ x\ and\ f(0)=2,\) then f(x) is equal to

  • A x sin x
  • B x cos x + sin x + 2
  • C x sin x+2
  • D x cos x+2

Question - 5

If \(f(x)=\int{x^2+sin^2\ x\over 1+x^2}.sec^2xdx\) and f(0)=0, then f(1) is equal to

  • A \(1-{\pi\over 4}\)
  • B \({\pi\over 4}-1\)
  • C \(tan\ 1-{\pi\over 4}\)
  • D None of the above

Question - 6

If \(\int{f(x)\over log\ sin\ x}dx\)=log log sin x, the f(x) is equal to

  • A sin x
  • B cos x
  • C log sin x
  • D cot x

Question - 7

\(\int{dx\over \sqrt{sin^3\ x.sin(x+\alpha)}}\) is equal to

  • A \(2\ cosec\ \alpha \sqrt{cos\ \alpha+sin\ \alpha.cot x}+C\)
  • B \(-2\ cosec\ \alpha \sqrt{cos\ \alpha+sin\ \alpha.cot x}+C\)
  • C \(cosec\ \alpha \sqrt{cos\ \alpha+sin\ \alpha.cot x}+C\)
  • D None of the above

Question - 8

\(\int{dx\over (x+1)^2\sqrt{x^2}+2x+2}\) is equal to

  • A \(-{\sqrt{x^2+2x+2}\over x+1}+c\)
  • B \(-{\sqrt{x^2+2x+1}\over (x+1)^2}+c\)
  • C \({-\sqrt{x^2+2x+2}\over x+1}+c\)
  • D None of these

Question - 9

\(\int{dx\over cos^3\ x.\sqrt{sin2x}}\) is equal to

  • A \(\sqrt2(\sqrt{cos\ x}+{1\over 5}tan^{5/2}x)+C\)
  • B \(\sqrt2(\sqrt{tan\ x}+{1\over 5}tan^{5/2}x)+C\)
  • C \(\sqrt2(\sqrt{tan\ x}+{1\over 5}tan^{5/2}x)+C\)
  • D \(\sqrt2(\sqrt{tan\ x}+{1\over 5}tan^{5/2}x)+C\)

Question - 10

\(\int{dx\over (1+\sqrt{x})\sqrt{x-x^2}}\) is equal to

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