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 ${1-\sqrt x\over 2\sqrt{1-x}}+C$