Indefinite Integration

If $y = y(x)$ satisfies the differential equation

$$8\sqrt{x}\left(\sqrt{9 + \sqrt{x}}\right)dy = \left(\sqrt{4 + \sqrt{9 + \sqrt{x}}}\right)^{-1}dx, \quad x > 0$$

and $y(0) = \sqrt{7}$, then $y(256) =$

Let $I_n=\displaystyle\int\tan^nx\,dx$, $(n>1)$. If $I_4+I_6=a\tan^5x+bx^5+C$, where C is a constant of integration, then the ordered pair (a, b) is equal to :

The integral $\displaystyle\int\sqrt{1+2\cot x(\operatorname{cosec}x+\cot x)}\,dx$ $\left(0<x<\dfrac\pi2\right)$ is equal to :

(where C is a constant of integration)

The integral $\displaystyle\int_{\pi/12}^{\pi/4}\frac{8\cos2x}{(\tan x+\cot x)^3}\,dx$ equals :

If $f\left(\dfrac{3x-4}{3x+4}\right)=x+2$, $x\ne-\dfrac43$, and $\displaystyle\int f(x)\,dx=A\log|1-x|+Bx+C$, then the ordered pair (A, B) is equal to :

(where C is a constant of integration)

If $\displaystyle\int_1^2\frac{dx}{(x^2-2x+4)^{3/2}}=\frac k{k+5}$, then k is equal to :

The value of $\displaystyle\int_{-\pi/2}^{\pi/2}\frac{x^2\cos x}{1+e^x}\,dx$ is equal to

The integral $\displaystyle\int\frac{2x^{12}+5x^9}{\left(x^5+x^3+1\right)^3}\,dx$ is equal to :

where C is an arbitrary constant.

If $\displaystyle\int\frac{dx}{\cos^3x\sqrt{2\sin2x}}=(\tan x)^A+C(\tan x)^B+k$, where k is a constant of integration, then A + B + C equals :

The integral $\displaystyle\int\frac{dx}{(1+\sqrt x)\sqrt{x-x^2}}$ is equal to :

(where C is a constant of integration.)

If
$$\alpha=\int_0^1\left(e^{9x+3\tan^{-1}x}\right)\left(\frac{12+9x^2}{1+x^2}\right)dx$$
where $\tan^{-1}x$ takes only principal values, then the value of $\left(\log_e|1+\alpha|-\dfrac{3\pi}4\right)$ is

Let $f(x)=7\tan^8x+7\tan^6x-3\tan^4x-3\tan^2x$ for all $x\in\left(-\dfrac\pi2,\dfrac\pi2\right)$. Then the correct expression(s) is(are)

If $\displaystyle\int_1^3x^2F'(x)\,dx=-12$ and $\displaystyle\int_1^3x^3F''(x)\,dx=40$, then the correct expression(s) is(are)

The integral $\displaystyle\int\frac{dx}{x^2(x^4+1)^{3/4}}$ equals :

The integral $\displaystyle\int\frac{dx}{(x+1)^{\frac34}(x-2)^{\frac54}}$ is equal to :

If $\displaystyle\int\frac{\log\left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}}dt=\frac12(g(t))^2+C$, where C is a constant, then g(2) is equal to :

The value of
$$\int_0^14x^3\left\{\frac{d^2}{dx^2}(1-x^2)^5\right\}dx$$
is

The following integral
$$\int_{\pi/4}^{\pi/2}(2\operatorname{cosec}x)^{17}\,dx$$
is equal to

The value of $g\left(\frac12\right)$ is

The integral $\displaystyle\int\left(1+x-\frac1x\right)e^{x+\frac1x}\,dx$ is equal to :