Definite Integration Questions (1340)

Assertion (A): If the primitive of \(f(x) = \sin x + 2x - 4\) has the value 3 for \(x = 1\), then there are exactly two values of \(x\) for which the primitive of \(f(x)\) vanishes.Reason (R): \(\cos x\) has period \(2\pi\).

If \int 2 2x +5x+9 dx = x\sqrtx 2 + x + 1 + \alpha\sqrtx 2 + x + 1+ \beta log ∣ e∣ x + 1 2 + \sqrtx 2 + x + 1∣ ∣ + C , where C is the \sqrtx2 +x+1 constant of integration, then \alpha + 2\beta is equal to _______.

Let \(f(x)\) be a function satisfying \(f'(x) = f(x)\) and \(f(0) = 2\). Then \(\int \frac{f(x)}{3 + 4f(x)} dx\) is

Evaluate $\int \frac{dx}{2x^2+x-1}$

If \(I = \int_{-1}^{1} \frac{\cos^{-1}\left(\frac{x^4}{4}\right)}{1+x^2} dx = k\int_{0}^{1} \frac{\cos^{-1}\left(\frac{x^4}{4}\right)}{1+x^2} dx\), then find \(k\).

If A(n) represents the area bounded by the curve y = n ln x, where n ∈ ℕ and n > 1, the x-axis and the lines x = 1 and x = e, then the value of A(n) + nA(n–1) is equal to:

\(\displaystyle\int\frac{dx}{(x-\alpha)(x-\beta)}\) equals \((\alpha\neq\beta)\)

Evaluate \(\int \frac{1 - x^2}{x(1 - 2x)} dx\)

Evaluate $\int \frac{dx}{\sqrt{(x-a)(b-x)}} (b > a)$

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:

∫ dx/∛(x⁵(x+1)⁷) is equal to:

The integral ∫ \frac{3x^2 + 1}{(x^2 - 1)^3} dx equals (where K is constant of integration)

Given \(I = \int \cos(\log_e x)\, dx\). Then \(I\) equals:

Let \(I = \int \frac{e^{4x} - e^{2x}}{e^{4x} + e^{2x} + 1} dx\) and \(J = \int \frac{e^{2x}}{e^{4x} + e^{2x} + 1} dx\). Then, for an arbitrary constant C, the value of J – I equals [IIT - 2008]

If $f(x) = \displaystyle\int \frac{1}{x^{1/4}(1+x^{1/4})}\,dx$, $f(0) = -6$, then $f(1)$ is equal to:

Let \(a \in (0, \pi/2)\) be fixed. If the integral \(\int \frac{\tan x - \tan a}{\tan x + \tan a} dx = A(x) \cos 2a + B(x) \sin 2a + C\), where \(C\) is a constant of integration, then the functions \(A(x)\) and \(B(x)\) are respectively

\(\int \frac{1+x\cos x}{x(1+x^2 e^{2\sin x})} dx\) is equal to

Evaluate ∫ (∛x + ∛(2-x²))(∛(1-x²) - ∛(2-x²))dx/∛(1-x³) for x ∈ (0,1):

13. ∫(sin(101x)·sin⁹⁹x)dx equals

Evaluate: \(\int \frac{dx}{(x + 2)(x^2 + 1)}\)

If \(\int \frac{\csc^2 x - 2010}{\cos^{2010} x} dx = -\frac{f(x)}{(g(x))^{2010}} + C\); where \(f\left(\frac{\pi}{4}\right) = 1\); then the number of solutions of the equation \(\frac{f(x)}{g(x)} = \{x\}\) in \([0, 2\pi]\) is/are: (where \(\{\}\) represents fractional part function)

If \(\displaystyle\int e^{\sin x}\left(\dfrac{x\cos^3 x - \sin x}{\cos^2 x}\right)dx = e^{\sin x}\big(f(x) - \sec x\big) + C\), then \(\dfrac{f(7)}{2}\) is equal to

Assertion (A): For \(-1 , the value of \((a + k)\) is \(\frac{9}{2}\).Reason (R): The given integral reduces to the form \(\int \frac{f'(x)}{f(x)} dx\) where \(f(x) = (x-1)^{5/2}\).

Let $I(x)=\displaystyle\int\sqrt{\dfrac{x+7}{x}}\,dx$ and $I(9)=12+7\ln 7$. If $I(1)=\alpha+7\ln(1+2\sqrt{2})$, then $\alpha^4$ is equal to _____.

If \int e ( x x sin -1 x + sin -1 x + x ) dx = g(x) + C , where C is the constant of integration, then g ( 1 ) equals : 3/2 1-x 2 2 \sqrt1-x2 (1-x ) 2

Evaluate : $\int \frac{2+3\cos\theta}{\sin\theta+2\cos\theta+3} d\theta$

Evaluate: \(\int \frac{\cos ec^2 x - 2005}{\cot x + \tan x} dx\)

∫dx(x-α)√((x-α)(x-β))equals

Let \int x sin x dx = g(x) + C , where C is the constant of integration. If 3 8 (g ( \pi 2 ) + g ( ′ \pi 2 )) = \alpha\pi 3 + \beta\pi 2 + \gamma, \alpha, \beta, \gamma \in Z , then \alpha + \beta - \gamma equals :

\(\int \frac{x^2 - 1}{(x^4 + 3x^2 + 1)\tan^{-1}\left(x + \frac{1}{x}\right)} dx\) is equal to

Suppose J = ∫ \frac{\sin^2 x + \sin x}{1 + \sin x + \cos x} dx and K = ∫ \frac{\cos^2 x + \cos x}{1 + \sin x + \cos x} dx. If C is an arbitrary constant of integration then which of the following is/are correct?

The integral $\displaystyle\int\dfrac{(x^8-x^2)\,dx}{(x^{12}+3x^6+1)\tan^{-1}\!\left(x^3+\dfrac{1}{x^3}\right)}$ is equal to:

Primitive of \(\dfrac{3x+1}{(x+1)^2\sqrt{x}}\) w.r.t. \(x\) is

Evaluate\[ I = \int \sec^{2/3} x\, \csc^{4/3} x\, dx \]

78. \(\int |x| dx\) is equal to

Let $\displaystyle\int x^3\sin x\,dx = g(x)+C$, where $C$ is the constant of integration. If $8\!\left(g\!\left(\dfrac{\pi}{2}\right)+g'\!\left(\dfrac{\pi}{2}\right)\right) = \alpha\pi^3+\beta\pi^2+\gamma$, $\alpha,\beta,\gamma\in\mathbb{Z}$, then $\alpha+\beta-\gamma$ equals:

∫ dx/√(1-tan²x) = λ sin⁻¹(λ sinx) + C, then λ = ?

Let $f(x)=\displaystyle\int\dfrac{dx}{(3+4x^2)\sqrt{4-3x^2}}$, $|x|<\dfrac{2}{\sqrt{3}}$. If $f(0)=0$ and $f(1)=\dfrac{1}{\alpha}\tan^{-1}\!\left(\dfrac{\alpha}{\beta}\right)$, $\alpha,\beta>0$, then $\alpha^2+\beta^2$ is equal to _______.

Assertion (A): The function \(F(x)\) (an indefinite integral of \(\sin 2x\)) satisfies \(F(x + \pi) = F(x)\) for all real \(x\).Reason (R): \(\sin 2(x + \pi) = \sin 2x\) for all real \(x\).

Evaluate \(\int \frac{e^x(1+x)}{\cos^2(e^x x)} dx\)

Evaluate \(\int \sqrt{x^2 + 4x + 1}\,dx\)

\(\int (\sin 2x - \cos 2x) \, dx = \frac{1}{2}\sin(2x - a) + b\), then

7. Integral of $\sqrt{1+2\cot x(\cot x+\csc x)}$ w.r.t. $x$ is

The value of $\int \frac{\ln\left(\frac{x-1}{x+1}\right)}{x^2 - 1} dx$ is equal to

If $f\left(\frac{1-x}{1+x}\right)=x$ and $g(x)=\int f(x)dx$ then

Let g(x) be an antiderivative for f(x). Then ln(1+(g(x))^2) is an antiderivative for

If $f\left(\frac{1-x}{1+x}\right)=x$ and $g(x)=\int f(x)dx$ then(A) $g(x)$ is continuous in domain(B) $g(x)$ is discontinuous at two points in its domain(C) $\lim_{x\to\infty} g'(x)=-1$(D) $\int g(x)dx=-\frac{x^2}{2}+(2x+1)\ln\left(\frac{1+x}{e}\right)+C$

The integral ∫ sec^2 x / (sec x + tan x)^9/2 dx equals (for some arbitrary constant K)

The value of $\int \frac{\ln\left(\frac{x-1}{x+1}\right)}{x^2-1} dx$ is equal to

10.Column-IColumn-II(A) Let $f(x) = \int x^{\sin x} (1 + \cos x \ln x + \sin x) dx$ and $f\left(\frac{\pi}{2}\right) = \frac{\pi^2}{4}$ then the value of $f(\pi)$ is(P) rational(B) Let $g(x) = \int \frac{1 + 2 \cos x}{(\cos x + 2)^2} dx$ and $g(0) = 0$ then the value of $g\left(\frac{\pi}{2}\right)$ is(Q) irrational(C) Let $k(x) = \int \frac{(x^2 + 1) dx}{\sqrt[3]{x^3 + 3x + 6}}$ and $k(-1) = \frac{1}{\sqrt{2}}$ then the value of $k(-2)$ is(R) integral(D) If $\int \frac{\cos x - \sin x + 1 - x}{e^x + \sin x + x} dx = \ln(f(x)) + g(x) + C$ (where $C$ is the constant of integration and $f(x)$ is positive), then $f(0) + g(0)$ is(S) prime