StudyDojo

Question 1

If $\int {{{({x^2} + 1){e^x}} \over {{{(x + 1)}^2}}}dx = f(x){e^x} + C}$ , where C is a constant, then ${{{d^3}f} \over {d{x^3}}}$ at x = 1 is equal to :

Question 2

1.

$\lim_{x \rightarrow 0} \frac{48}{x^{4}} \int_{0}^{x} \frac{t^{3}}{t^{6}+1} \mathrm{~d} t$ is equal to ___________.

[4]

Question 3

The value of $\int\limits_{{{ - \pi } \over 2}}^{{\pi \over 2}} {{1 \over {1 + {e^{\sin x}}}}dx}$ is:

Question 4

If $2\int\limits_0^1 {{{\tan }^{ - 1}}xdx = \int\limits_0^1 {{{\cot }^{ - 1}}} } \left( {1 - x + {x^2}} \right)dx,$

then $\int\limits_0^1 {{{\tan }^{ - 1}}} \left( {1 - x + {x^2}} \right)dx$ is equalto :

Question 5

1.

Let $f_{n} = \int_{0}^{\frac{\pi}{2}} \left( \sum_{k=1}^{n} \sin^{k-1} x \right) \left( \sum_{k=1}^{n} (2k-1) \sin^{k-1} x \right) \cos x \, dx, \quad n \in \mathbb{N}$ . Then $f_{21} - f_{20}$ is equal to _________

[4]

Question 6

$\int\limits_{ - \pi }^\pi {\left| {\pi - \left| x \right|} \right|dx}$ is equal to :

Question 7

If $\int\limits_0^x \,$ f(t) dt = x² + $\int\limits_x^1 \,$ t²f(t) dt then f ' $\left( {{1 \over 2}} \right)$ is -

Question 8

The integral $$ \int\left[\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right] \ln \left(\frac{e x}{2}\right) d x

Question 9

$\text { The integral } \int \frac{\left(x^8-x^2\right) \mathrm{d} x}{\left(x^{12}+3 x^6+1\right) \tan ^{-1}\left(x^3+\frac{1}{x^3}\right)} \text { is equal to : }$

Question 10

The integral $\int \cos(\log_e x) \, dx$ is equal to : (where C is a constant of integration)

Calculus

Calculus