Integral Calculus Questions (265)

Let $P(x)$ be a quadratic polynomial with $P(1)=-1$. If $\displaystyle\int\frac{P(x)\,dx}{(2x-3)^2(3x-2)^2}=\frac{1}{5}\ln|f(x)|+C$ where $f(x)$ is rational and $\lim_{x\to\infty}f(x)=\frac{4}{3}$, then $|f(1)|$ is

The value of the definite integral $\displaystyle\int_{-2}^{2}x^3\ln(1^x+3^x+5^x+15^x)\,dx$

Let $f$ be a function defined by $f(x)=\begin{cases}(x-r)^2, & r-1\leq x<r+1\\ 1, & r+1\leq x\leq r+2\end{cases}$ where $r=3k$, $k\in I$. Find $\displaystyle\sqrt{\int_0^{45}f(x)\,dx}$

If $\displaystyle\int\frac{\sec^2 x-2010}{\sin^{2010}x}\,dx=\frac{P(x)}{(\sin x)^{2010}}+C$, then the value of $P\!\left(\dfrac{\pi}{3}\right)$ is

The area bounded by the curve $y = \dfrac{1}{x^2 - 2x + 2}$ and the $x$-axis equals

The area enclosed between the curves $y=ax^2$ and $x=ay^2$ ($a>0$) is 1 sq. unit. Then the value of $a$ is

Let $I=\displaystyle\int_0^2\!\left[\left|x^2-5x+4\right|+\left[\sin\frac{3\pi}{2}x\right]\right]dx$ (where $[\cdot]$ is GIF). Then $I+\dfrac{2}{3}$ is

If $I=\displaystyle\int\frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}\,dx$, then $I$ equals

If $f'(x) = \frac{1}{-x + \sqrt{x^2 + 1}}$ and $f(0) = -\frac{1 + \sqrt{2}}{2}$, then the value of $f(5)$ will be :

The integral $\int \frac{\sec^{3/2}\theta - \sec^{1/2}\theta}{2 + \tan^2\theta} \tan\theta d\theta$ is:

If $f(x) = \lim_{n \to \infty} \frac{x^n - x^{-n}}{x^n + x^{-n}}$, $0 < x < 1$, $n \in \mathbb{N}$ then $\int (\sin^{-1} x) f'(x) dx$ is equal to:

The derivative of $x^4 + x^{-5}$ is $-\left(4x^{-5} + 5x^{-6}\right)$. So, $$\int \frac{5x^3 + 4x^5}{\left(x^5 + x + 1\right)^2} dx =$$

If the anti-derivative of $\frac{x^3}{\sqrt{4+2x^2}}$ which passes through $(1, 2)$ is $\frac{1}{m}\left(1+2x^2\right)^{1/2}\left(x^2-1\right)+c$. Then:

Let $f(xy) = f(x) \cdot f(y)$, $\forall x > 0, y > 0$ and $f(1 + x) = 1 + x[1 + g(x)]$, where $\lim_{x \to 0} g(x) = 0$, then $$\int \frac{f(x)}{f'(x)} dx$$ is:

If $$\int \frac{dx}{x^2(x^n + 1)^{(n-1)/n}} = -[f(x)]^{1/n} + C$$, then $f(x)$ is

Let $f(x) = \frac{1}{4 - 3\cos^2 x + 5\sin^2 x}$ and its anti-derivative $F(x) = \frac{1}{3}\tan^{-1}(g(x)) + c$, then :

$I_1 = \int f(x) dx$ and $I_2 = \int_0^1 f(x) dx$ where $f(x) = x^2 \ln\left(1-x^2\right)$, then:

$f(x) = \int e^{\tan^{-1}x}\left(1+x+x^2\right)d\left(\cot^{-1}x\right)$ is equal to:

If $\int\left[\frac{1}{1-x^8}\left\{\cos^{-1}\left(\frac{2x}{1+x^2}\right) + \tan^{-1}\left(\frac{2x}{1-x^2}\right)\right\}\right]dx$ has $p\tan^{-1}f(x)$ & $q\tan^{-1}g(x)$ terms and $x \in (-1, 1)$, then: (where $p$ and $q$ are constant)

If $A = \int e^{ax}\cos bx dx$ and $B = \int e^{ax}\sin bx dx$, then which of the following may be correct?

If $\int \sqrt{\frac{\cos x - \cos^3 x}{\left(1-\cos^3 x\right)}} dx = f(x) + c$, then $f(x)$ is equal to:

Let $f$ is a differentiable function such that $f(x) = x^2 + \int_0^x e^{-t}f(x-t)dt$, then:

Let $f$ is a differentiable function such that $f'(x) = f(x) + \int_0^2 f(x)dx, f(0) = \frac{4 - e^2}{3}$, then:

The value of $\int_1^a \frac{x^a - 1}{\log x} dx$ is:

The value of $\int_0^{\pi/2} \log(\sin^2 \theta + k^2 \cos^2 \theta) d\theta$, where $k \geq 0$, is:

The value of $\frac{dI}{da}$ when $I = \int_0^{\pi/2} \log\left(\frac{1 + a \sin x}{1 - a \sin x}\right) \frac{dx}{\sin x}$ (where $|a| < 1$) is:

If $p, q, r, s$ are in arithmetic progression and $f(x) = \begin{vmatrix} p + \sin x & q + \sin x & p - r + \sin x \\ q + \sin x & r + \sin x & -1 + \sin x \\ r + \sin x & s + \sin x & s - q + \sin x \end{vmatrix}$ such that $\int_0^2 f(x) dx = -4$, then the common difference of the progression is:

If $\int_0^1 \frac{\sin t}{1+t} dt = a$, then the value of $\int_{4\pi-2}^{4\pi} \frac{\sin t}{4\pi + 2 - t} dt$ is:

Least positive value of $c$ if $c, k, b$ are in A.P. is:

If $G(x,t) = \begin{cases} x(t-1), & \text{when } x \leq t \\ t(x-1), & \text{when } t < x \end{cases}$ and if $f$ is continuous function of $x$ in $[0,1]$. Let $g(x) = \int_0^1 f(t)G(x,t)dt$. Then which is incorrect:

If $a \leq \int_0^1 \frac{dx}{\sqrt{4-x^2-x^3}} \leq b$, then $(a,b) =$

If $P = \int_0^\infty \frac{x^2}{1+x^4} dx$; $Q = \int_0^\infty \frac{xdx}{1+x^4}$ and $R = \int_0^\infty \frac{dx}{1+x^4}$, then:

Let $u = \int_0^{\pi/4} \left(\frac{\cos x}{\sin x + \cos x}\right)^2 dx$ and $v = \int_0^{\pi/4} \left(\frac{\sin x + \cos x}{\cos x}\right)^2 dx$, then:

Let $g(x) = x^c e^{2x}$ & let $f(x) = \int_0^x e^{2t}(3t^2+1)^{1/2} dt$. For a certain value of 'c', the limit of $\frac{f'(x)}{g'(x)}$ as $x \to \infty$ is finite and non-zero, then:

If $\int_0^2 \frac{\ln(1+2x)}{1+x^2} dx = (\tan^{-1}a)(\ln\sqrt{b})$ where $a,b \in \mathbb{N}$, then:

Which of the following is/are true?

Let $f$ be continuous and differentiable in $(x_1, x_2)$. If $f(x)f'(x) \geq x\sqrt{1-[f(x)]^4}$ and $\lim_{x\to x_1}(f(x))^2=1$, $\lim_{x\to x_2}(f(x))^2=\frac{1}{2}$. Then minimum value of $\left[x_1^2 - x_2^2\right]$ is ........... (where $[\cdot]$ denotes GIF)

The integral $\int \frac{\sec^{3/2}\theta - \sec^{1/2}\theta}{2 + \tan^2\theta} \tan\theta d\theta$ is:

If $f(x) = \lim_{n \to \infty} \frac{x^n - x^{-n}}{x^n + x^{-n}}$, $0 < x < 1$, $n \in \mathbb{N}$ then $\int (\sin^{-1} x) f'(x) dx$ is equal to:

$$\int \frac{x^4 - 2}{x^2\sqrt{x^4 + x^2 + 2}} dx =$$

Anti-derivative of $$\frac{x - 1}{(x + 1)\sqrt{x^3 + x^2 + x}}$$ is:

The derivative of $x^4 + x^{-5}$ is $-\left(4x^{-5} + 5x^{-6}\right)$. So, $$\int \frac{5x^3 + 4x^5}{\left(x^5 + x + 1\right)^2} dx =$$

If $f'(x) = \frac{1}{-x + \sqrt{x^2 + 1}}$ and $f(0) = -\frac{1 + \sqrt{2}}{2}$, then the value of $f(5)$ will be :

Let $f(xy) = f(x) \cdot f(y)$, $\forall x > 0, y > 0$ and $f(1 + x) = 1 + x[1 + g(x)]$, where $\lim_{x \to 0} g(x) = 0$, then $$\int \frac{f(x)}{f'(x)} dx$$ is:

If $$\int \frac{dx}{x^2(x^n + 1)^{(n-1)/n}} = -[f(x)]^{1/n} + C$$, then $f(x)$ is

If $I_{m,n} = \int \cos^m x \sin nx dx$, then $7I_{4,3} - 4I_{3,2} =$$

$I = \int \frac{dx}{(\sin x - 2\cos x)(2\cos x + \sin x)}$ is equal to

If $\int \sqrt{\cos ecx + 1}\,dx = kfog(x) + c$, where $k$ is a real constant, then :

Let $f(x) = \frac{1}{4 - 3\cos^2 x + 5\sin^2 x}$ and its anti-derivative $F(x) = \frac{1}{3}\tan^{-1}(g(x)) + c$, then :

A function $f(x)$ continuous on $\mathbb{R}$ and periodic with $2\pi$ satisfies $f(x) + (\sin x) f(x + \pi) = \sin^2 x$ then,