Integrals - Class 12

Let $S$ be the set of all twice continuously differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(0) = 3$ and $\int_0^x \begin{vmatrix} f(t) & \sin t \\ f''(t) & -\sin t \end{vmatrix} dt = f(x) \cos x - 3 - \sin x (f'(x))^3$ for all $x \in \mathbb{R}$. Find the value of $\sum_{f \in S} f\left(\frac{\pi}{2}\right)$.

Let $S_n(x) = \frac{1369}{2} \left( n - \frac{\sin(nx)\cos((n+1)x)}{\sin x} \right)$ for $x \neq m\pi$, where $m \in \mathbb{Z}$. A sequence of functions is defined as $T_n(x) = S_n(x) - S_{n-1}(x)$ for $n \ge 2$, and $U_n(x) = \sqrt{T_n(x)}$. Evaluate the integral $I = \int_0^{\pi/4} U_{10}(x) dx$

A continuous random variable $X$ has a probability density function $f(x) = \frac{1}{2} f_1(x) + \frac{1}{2} f_2(x)$, where $f_1(x) = 1$ and $f_2(x) = 2x$ for $x \in [0, 1]$, and $f(x) = 0$ otherwise. Let $A$ be the event $X \in \left[0, \frac{1}{2}\right]$ and $B$ be the event $X \in \left[\frac{1}{4}, \frac{3}{4}\right]$. What is the conditional expectation $E[X \mid A \cup B]$?

Let $A(x) = \begin{bmatrix} 1 & 1 & 1 \\ 1 & x & x^2 \\ 2x & 3x^2 & 4x^3 \end{bmatrix}$. Let $I_1 = \int_0^1 \det(A(x)) \, dx$ and $I_2 = \det\left( \int_0^1 A(x) \, dx \right)$. Find the value of $900(I_1 + I_2)$.

Evaluate the integral: $I = \int_{0}^{\sqrt{3}} x \left( \arcsin\left(\frac{2x}{1+x^2}\right) - \arccos\left(\frac{1-x^2}{1+x^2}\right) \right) dx$

Let $M(x)$ be a $3 \times 3$ matrix defined as: $M(x) = \begin{pmatrix} 2e^x + e^{-x} + 1 & 2e^{2x} + e^{-2x} + 1 & 2e^{3x} + e^{-3x} + 1 \\ e^x + 2e^{-x} + 1 & e^{2x} + 2e^{-2x} + 1 & e^{3x} + 2e^{-3x} + 1 \\ e^x + e^{-x} + 2 & e^{2x} + e^{-2x} + 2 & e^{3x} + e^{-3x} + 2 \end{pmatrix}$ Evaluate the integral: $I = \int_{-\ln 2}^{\ln 2} \frac{e^{2x} |\det(M(x))|}{4(e^x + 1)} \, dx$

Let $c_k$ be the coefficient of $x^k$ in the expansion of $x(1-x)^{-2}$ for $|x| < 1$. Define the function $f(t)$ for $t \neq 0$ as: $f(t) = \lim_{n \to \infty} \sum_{k=1}^{n} \frac{n(2n^2 - t^2 c_{k-1})}{(n^2 + t^2 c_k^2)(n^2 + t^2 c_{k-1}^2)}$ Evaluate the limit: $L = \lim_{x \to 0} \frac{1}{1 - \cos x} \int_{0}^{x} t \cdot f(t) dt$

Let $y = f(x)$ be the continuous real-valued function defined on $[-2, 2]$ satisfying the implicit equation $y^3 - 3y + x = 0$ and $f(0) = 0$. Evaluate the integral: $I = \int_{-2}^{2} \frac{f(x)}{\sqrt{2-x}} \, dx$

Let $P(\theta) = (\sin\theta, \cos\theta, \sin 2\theta)$ be a moving point in 3D space and $\Pi$ be the plane $x + y - z = 1$. If $d(\theta)$ denotes the perpendicular distance from $P(\theta)$ to the plane $\Pi$, evaluate the integral $I = \int_{-\pi/2}^{0} \sqrt{3} \, d(\theta) (\cos\theta - \sin\theta) d\theta$

Evaluate the integral: $I = \int_{-3\pi}^{3\pi} \frac{|x \cos x|}{1 + 2024^x} \, dx$

Let $f(t) = \pi \sin(\pi t)$, $g(t) = e^t$, and $h(t) = 12t^2$. Define the function $\Delta(x)$ as the determinant: $\Delta(x) = \begin{vmatrix} \int_0^x f(t) dt & \int_0^{x^2} g(t) dt & \int_0^{x^3} h(t) dt \\ x & x^2 & x^3 \\ 1 & 2x & 3x^2 \end{vmatrix}$ Evaluate the integral $I = \int_0^1 x \Delta''(x) dx$.

Let $y = y(x)$ be the continuous solution to the differential equation $\frac{dy}{dx} + \text{sgn}(x) y = |x| \cos(x) e^{-|x|}$ for all $x \in \mathbb{R}$, subject to the condition $y(\pi) = e^{-\pi}$. Find the value of $y(-\pi)$.

Let $P(x) = \frac{1}{720} \sum_{k=0}^{5} (-1)^k \binom{5}{k} (x-k)^6$. Evaluate the integral $\int_{1}^{e} P(x) \ln x \, dx$

Let $f(x) = \\sum_{k=1}^{\\infty} \\frac{2}{(x+k)(x+k+1)(x+k+2)}$. Define $H(x) = \\int_0^x f(t) dt - \\int_0^x \\frac{1}{2(1+t^3)} dt$. The number of positive real roots of the equation $H(x) = 0$ is:

Let $y = y(x)$ be a differentiable function on $\mathbb{R}$ satisfying the differential equation $(x-2) \frac{dy}{dx} - y = \frac{(x-2)^7 + 313(x-2)^5 + 21(x-2)^3}{(x-2)^4 + 4(x-2)^2 + 5} - 1$ If $y(3) = 2024$, then the value of $y(4) + y(0)$ is

Evaluate the integral $I = \int_0^{3\pi} \frac{1 + 2\sin x - \cos x}{5 + 3\cos x} \, dx$

Let $A_r = \sum_{k=0}^r \binom{n}{k} \binom{n-k}{r-k}$ for $0 \le r \le n$. For an integer $n \ge 2$, define the function $f_n(x)$ for $x > -\frac{1}{2}$ by $f_n(x) = \int_0^x \frac{\sum_{r=2}^n A_r t^r}{t^2(2t+1)} dt$ Let $g_n(x) = f_n(x) - 2(n-1)\ln(2x+1)$. Let $S$ be the set of all integers $n \ge 2$ such that $g_n(x)$ is a constant function on $\left(-\frac{1}{2}, \infty\right)$. Find the value of $\sum_{n \in S} n + g_4(1)$.

Let $H$ be the rectangular hyperbola $x^2 - y^2 = 2$. A curve $C$ is obtained by rotating $H$ about the origin by an angle of $\pi/4$ clockwise. Let $P(x, y)$ be a point on $C$ in the first quadrant with $x \ge 1$, and let $D(x)$ be the distance from $P$ to the origin. Define the integral: $I(\lambda) = \int_{1}^{\infty} \frac{x^2 - 1}{x^2 D(x)^2 + (\lambda - 2)x^2} \, dx$ Using the method of differentials, the approximate value of $I(4.48)$ is:

Let $f(\theta) = \int_0^{\pi/2} \ln(1 - \cos^2\theta \sin^2 x) \, dx$ for $\theta \in (0, 2\pi)$. Find the value of $f''\left(\frac{7\pi}{6}\right)$.

Let $\omega$ be a complex cube root of unity ($\omega \neq 1$). Evaluate the following limit: $\lim_{x \to 0} \frac{ \sin^2 x + \omega^2 x \sin x + \omega x^2 }{ \left( \cos^2 x + \omega^2 \cos x + \omega \right) \left( \frac{\sin(2x)\cos(5x/2)}{\sin(x/2)} - 4 \right) }$

Let $f(x) = 16 \sin^4(2x) \left( \sqrt{\cos^4 x - \sin^4 x - 1} \right)^2$ be a real-valued function. Evaluate the definite integral: $I = \int_{0}^{\pi/2} f(x) \, dx$

Let $f(x) = \int_{0}^{\sin x} \left( \arcsin t - \frac{\pi}{6} \right) dt$. Find the value of $\int_{0}^{2\pi} |f'(x)| dx$.

Let $S$ be the set of all real values of $k$ for which the quadratic equations $x^2 - kx - 5 = 0$ and $x^2 + 5x + k = 0$ have at least one common root. Consider the integral $I(k) = \int_{-\infty}^{\infty} \frac{dx}{x^2 + 2kx + k^3 - 6k^2 - 14k + 120}$ If $k \in S$ is chosen such that $I(k)$ converges to a finite real number, find the value of $I(k)$.

Let $D$ be the maximal subset of $\mathbb{R}$ for which the function $f(x) = \sqrt{5 + 4x - x^2} - \sqrt{x^2 - 6x + 9}$ is real-valued. The value of the integral $\int_{D} x f(x) \,dx$ is

Let $I = \int_{1}^{15} \left( \sum_{k=1}^{[x]} \frac{k^2 \sin^k \left( \frac{\pi \{x\}}{2} \right)}{ \sin^k \left( \frac{\pi \{x\}}{2} \right) + \cos^k \left( \frac{\pi \{x\}}{2} \right) } \right) dx$ where $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}$ denotes the fractional part of $x$. The value of $I$ is:

Let $A(x)$ be a $3 \times 3$ real matrix whose entries are continuous functions of $x$. Suppose $A(x) A(x)^T = I$ and $\det(A(x)) = 1$ for all $x \in \mathbb{R}$. If the characteristic polynomial of $A(x)$ is given by $P(\lambda, x) = \lambda^3 - \alpha(x)\lambda^2 + \beta(x)\lambda - \gamma(x)$, evaluate the integral: $I = \int_{-\pi/2}^{\pi/2} \frac{\alpha(x) - \beta(x) + \gamma(x) + x^2 \cos x}{1 + 2024^x} \, dx$

Let $z = e^{ix}$. Evaluate the definite integral: $I = \int_{0}^{\pi} \text{Im} \left( \text{Log} \left( z + z^2 \right) \right) dx$ where $\text{Log}(w)$ denotes the principal branch of the complex logarithm, such that $\text{Im}(\text{Log}(w)) \in (-\pi, \pi]$.

Let $I = \int_{0}^{\pi} x \sin^2 x \left[ \frac{2x}{\pi} \right] dx$, where $[t]$ denotes the greatest integer less than or equal to $t$. The value of $I$ is:

Let $A(x) = \begin{pmatrix} \frac{1}{2} & \sin^3(\pi x) & e^{x^2} \\\\ 0 & 1 & \ln(1+x^2) \\\\ 0 & 0 & 1 \end{pmatrix}$ and $B(x) = \begin{pmatrix} 1 & 1 & 1 \\\\ x^2 & 1 & 3 \\\\ x^4 & 1 & 9 \end{pmatrix}$.

If $M(x) = A(x)B(x)$, evaluate the integral: $I = \int_{-\sqrt{3}}^{\sqrt{3}} \frac{|\det(M(x))|}{1 + 2024^x} \, dx$

Let $A(t) = \begin{pmatrix} \ln\left(\frac{2-t}{2+t}\right) & \frac{\pi}{4} \cos\left(\frac{\pi t}{2}\right) \\ |t|^3 + |t| & -\ln\left(\frac{2-t}{2+t}\right) \end{pmatrix}$ for $t \in (-2, 2)$. Let $M(x) = \int_{-x}^{x} A(t) dt$ and define the function $I(x) = \text{tr}(M(x)^4)$. The value of $I'(1)$ is:

Let $F(x) = \int_{0}^{x} x^4 \left( \sum_{k=1}^{\infty} \left( \frac{t^2}{x^2+t^2} \right)^k \right) \left( \sum_{n=1}^{\infty} \frac{1}{(t^2 + n x^2)(t^2 + (n+1) x^2)} \right) dt$ for $x > 0$. Find the value of $F'(2) + \frac{\pi}{4}$.

Let $f(x) = \int_{0}^{\pi} \frac{\sin^2 t - \sin^2 x}{|\sin t - \sin x|} dt$ for $x \in \left(0, \frac{\pi}{2}\right)$. Find the value of $f'\left(\frac{\pi}{4}\right)$.

Let $f(x) = \begin{vmatrix} \int_0^x \sin t \, dt & \int_0^x \cos t \, dt & \int_0^x e^{t^2} \, dt \\ \sin x & \cos x & e^{x^2} \\ \cos x & -\sin x & 2x e^{x^2} \end{vmatrix}$. Find the value of $\lim_{x \to 0} \frac{f(x)}{x^3}$.

Let $\Delta(x) = \begin{vmatrix} |x| - x & \frac{x}{\sqrt{x^2+1}} & \frac{1}{\sqrt{x^2+1}} \\ -\frac{x}{\sqrt{x^2+1}} & |x| - x & \tan x \\ -\frac{1}{\sqrt{x^2+1}} & -\tan x & |x| - x \end{vmatrix}$. Evaluate the definite integral $\int_{-\pi/3}^{\pi/3} \Delta(x) \, dx$.

Let $c_k$ be the term independent of $t$ in the binomial expansion of $\left( t^2 + \frac{\sqrt{2}}{t} \right)^{3k}$. Define the polynomial $f(x) = \sum_{k=0}^{3} c_k x^{2k}$.

Evaluate the integral: $I = \int_{-1/2}^{1/2} \left( \frac{f(x)}{1 + e^{x^3 + \sin x}} + x^4 \ln\left(\frac{1-x}{1+x}\right) \right) dx$

Let $y = f(x)$ be a continuous function defined on $[-1, 1]$ which satisfies the differential equation $x(x^2 + y^2) dx + y(x^2 - y^2) dy = 0$ with the initial condition $f(0) = 1$. Evaluate the definite integral: $I = \int_{-1}^{1} \left( x^2 \ln\left( \frac{f(x) - x}{f(x) + x} \right) + \frac{3x^2 \left( (f(x))^2 - x^2 \right)}{\sqrt{2x^4+1}} \right) dx$

Let $\alpha$ and $\beta$ be the non-real roots of the equation $x^3 - 5x^2 + 11x - 15 = 0$. Evaluate the integral: $I = \int_{-3}^{7} \left[ \cot^{-1}\left( \frac{x - (\alpha+\beta)}{\sqrt{\alpha\beta - 1}} \right) + \cot^{-1}\left( \frac{\sqrt{\alpha\beta - 1}}{x - (\alpha+\beta)} \right) \right] dx$

Let $I_n = \int_0^1 x^n e^{x^2} \, dx$ for $n \in \mathbb{N} \cup \{0\}$. Find the value of $\sum_{n=1}^{50} \left[ (2n-1) I_{2n-2} \right]$ where $[x]$ denotes the greatest integer less than or equal to $x$.

Evaluate the definite integral: $I = \int_0^{\frac{21\pi}{2}} x \sqrt{1-\cos 2x} \cos^4 x \, dx$

Let $I(x) = \int_{\pi/2}^x \cos^3 t \, dt$. A curve $C$ is defined by $y = f(x)$ for $x \in (\pi/2, \pi)$, which satisfies the differential equation: $\frac{dy}{dx} + \frac{3 \cos^3 x}{|I(x)|} y = 4 \cos^3 x |I(x)|$ Given that $y(3\pi/4) = 4 \left( I(3\pi/4) \right)^2$. A particle moves along the curve $C$. At the instant when $x = 3\pi/4$, its $x$-coordinate is increasing at a rate of $\sqrt{2}$ units/sec. Find the rate of change of its $y$-coordinate at this instant.

Let $I_n = \int_0^{\pi/2} \Delta_n(x) \, dx$ for $n \ge 1$, where $\Delta_n(x) = \begin{vmatrix} \sin^n x & \sin^{n+2} x & \sin^{n+4} x \\ \sin^n x & \sin^n x \cos^2 x & \sin^n x \cos^4 x \\ 1 & -\sin^2 x & \sin^4 x \end{vmatrix}$ Which of the following recurrence relations holds true?

Let $I_n = \int_{-1}^{3} (x^n + n x^{n-1}) e^x \text{sgn}(x-1) \, dx$ for $n \in \mathbb{N}$, where $\text{sgn}(\cdot)$ denotes the signum function. Find the value of $\sum_{n=1}^{5} \frac{I_n - (-1)^n e^{-1}}{3^n}$.

Let $I = \sum_{k=0}^{6} \binom{6}{k} (-1)^k \int_{0}^{\pi/8} \sin^{2k} x \cos^{12-2k} x \, dx$ The value of $I$ is:

The family of curves $C_1$ satisfies the differential equation $\frac{dy}{dx} = \frac{\cos x}{y \sin x - \sin^4 x \cos^8 x}$. Let $C_2$ be the orthogonal trajectory to $C_1$ that passes through the point $(\pi, 0)$. If the area of the region bounded by the curve $C_2$, the x-axis, and the line $x = \frac{3\pi}{2}$ is $A$, then $A$ equals:

Evaluate the following limit: $\lim_{n \to \infty} \frac{1}{n^2} \sum_{r=n}^{2n-1} r \frac{\binom{4n}{r}}{\binom{4n}{r+1}}$

A discrete random variable $X_n$ takes values $k \in \{1, 2, \dots, 2n\}$. A transformed random variable is defined as $U_n = \frac{X_n - n}{n}$. The probability mass function of $X_n$ is given by $P(X_n = k) = \frac{1}{Z_n} (u_k + 1)^2 \cos\left( \frac{\pi}{2} u_k \right)$ where $u_k = \frac{k - n}{n}$ and $Z_n$ is the normalization constant. Evaluate $\lim_{n \to \infty} E[U_n]$.

Let $A_n = \left\{ k \in \mathbb{N} : 1 \le k \le 5n \text{ and } \left| \frac{2k}{n} - 3 \right| \le \frac{k}{n} + 1 \right\}$. Find the value of: $\lim_{n \to \infty} \frac{1}{n} \sum_{k \in A_n} 2^{\frac{3k}{n}}$

Let $A_k$ be a $5 \times 5$ matrix whose entries are given by $a_{ij} = \int_0^{k/n} t^{i+j} \, dt + \tan^{-1}\left( \frac{k}{n} (i-j) \right)$ for $1 \le i, j \le 5$. If $B_k = A_k - A_k^T$, then the value of $\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \left( \det(B_k) + \frac{k}{n} e^{(k/n)^2} \right)$ is equal to:

By bounding the oscillatory term $\sin^2(\pi x)$ by its minimum and maximum values, the strict bounds obtained for the integral $I = \int_0^2 \frac{|x^2 - x|}{2 + [x] + \sin^2(\pi x)} dx$ (where $[x]$ denotes the greatest integer function) are:

Consider the definite integrals: $I_1 = \left| \int_0^{2\pi} \left( e^{i \arcsin(\sin x)} - e^{i \arccos(\cos x)} \right) dx \right|$ $I_2 = \int_0^{2\pi} \left| e^{i \arcsin(\sin x)} - e^{i \arccos(\cos x)} \right| dx$ where $i = \sqrt{-1}$. Find the ordered pair $(I_1, I_2)$.