Let $$f:[-1,2] \rightarrow[0, \infty)$$ be a continuous function such that $$\mathrm{f}(x)=\mathrm{f}(1-x), \forall x \in[-1,2]$$ Let $$\mathrm{R}_1=\...

$$\int\limits_0^\pi \frac{d x}{4+3 \cos x}=$$

Let $$\mathrm{f}(x)=\int \frac{\sqrt{x}}{(1+x)^2} \mathrm{~d} x, x \geq 0$$, then $$\mathrm{f}(3)-\mathrm{f}(1)$$ is equal to

Let $$\mathrm{f}(x)$$ be positive for all real $$x$$. If $$\mathrm{I}_1=\int_\limits{1-\mathrm{h}}^{\mathrm{h}} x \mathrm{f}(x(1-x)) \mathrm{d} x$$ an...

$$\int_\limits{-1}^3\left(\cot ^{-1}\left(\frac{x}{x^2+1}\right)+\cot ^{-1}\left(\frac{x^2+1}{x}\right)\right) \mathrm{d} x=$$

Let $$\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$$ and $$\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$$ be continuous functions. Then the value of...

$$\int_\limits 0^\pi \frac{x \tan x}{\sec x+\cos x} d x= $$

$$\int_\limits0^1 \cos ^{-1} x d x=$$

If $$\int_\limits0^{\frac{1}{2}} \frac{x^2}{\left(1-x^2\right)^{\frac{3}{2}}} \mathrm{~d} x=\frac{\mathrm{k}}{6}$$, then the value of $$\mathrm{k}$$ i...

$$\int_\limits1^2 \frac{\mathrm{d} x}{\left(x^2-2 x+4\right)^{\frac{3}{2}}}=\frac{\mathrm{k}}{\mathrm{k}+5} \text {, then } \mathrm{k} \text { has the...

If $$\mathrm{f}(x)$$ is a function satisfying $$\mathrm{f}^{\prime}(x)=\mathrm{f}(x)$$ with $$\mathrm{f}(0)=1$$ and $$\mathrm{g}(x)$$ is a function th...

$$\int\limits_5^{10} \frac{d x}{(x-1)(x-2)}=$$

$$\int\limits_{{{ - \pi } \over 2}}^{{\pi \over 2}} {{{\cos x} \over {1 + {e^x}}}dx = } $$

$$\int_\limits0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1-\sin x \cos x} d x=$$

If $$f(x)=|x-1|+|x-2|+|x-3|, \forall x \in[1,4]$$, then $$\int_\limits1^4 f(x) d x=$$

If $$2 f(x)-3 f\left(\frac{1}{x}\right)=x$$, then $$\int_\limits1^e f(x) d x=$$

If $$\int_\limits2^e\left[\frac{1}{\log x}-\frac{1}{(\log x)^2}\right] d x=a+\frac{b}{\log 2}$$, then

$$\int_\limits0^{\pi / 4} \log (1+\tan x) d x=$$

If $$\int_\limits0^{\frac{\pi}{2}} \frac{d x}{5+4 \sin x}=A \tan ^{-1} B$$, then A + B =