Definite Integration Questions (1340)

∫ sin-1√xx+a dx equals

∫ \frac{5x^4 + 4x^5}{(x^5 + x + 1)^2} dx equals

Primitive of $\frac{3x^4 - 1}{(x^4 + x + 1)^2}$ w.r.t. $x$ is -

∫ \frac{5x^4 + 4x^5}{(x^5 + x + 1)^2} dx equals

79. \(\int \frac{(1-x^2)^{3/2}}{x^2} dx\) is equal to

Let $f(x)=\displaystyle\int\frac{dx}{x^{2/3}+2x^{1/2}}$ be such that $f(0)=-26+24\log_e 2$. If $f(1)=a+b\log_e 3$, where $a,b\in\mathbb{Z}$, then $a+b$ is equal to:

If ∫ 1 / ((x-1)^(4/3) * (x+3)^(5/3)) dx = A * ((αx - 1) / (βx + 3))^B + C, where C is the constant of integration, then the value of α + β + 20AB is ________.

For x ∈ (-π/2, π/2), if y(x) = ∫ (cosec x + sin x) / (cosec x sec x + tan x sin2 x) dx and lim x→π/2 y(x) = 0 then y(π/4) is equal to

If $I_n = \int (\sin x)^n dx, n \in N$, then $5I_4 - 6I_6$ is equal to -

MATRIX MATCH TYPE QUESTIONList-IList-II(P) $\int \frac{x^2(x^6+x^5-1)dx}{(2x^6+3x^5+2)^2}$(1) $-\frac{1}{3} \frac{1}{(x^3-x+1)} + C$(Q) $\int \frac{(x^5+x^4+x^2)dx}{\sqrt{4x^7+5x^6+10x^4}}$(2) $\frac{1}{2}(1+x^{-2}+x^{-5})^{-2} + C$(R) $\int \frac{(2x^{12}+5x^9)dx}{(x^5+x^3+1)^3}$(3) $-\frac{1}{6}(2x^3+3x^2+2x^{-3})^{-1} + C$(S) $\int \frac{x^2-\frac{1}{3}}{(x^3-x+1)^2} dx$(4) $x\left(\frac{x^3}{25} + \frac{x^2}{20} + \frac{1}{10}\right)^{\frac{1}{2}} + C$

For x ∈ (-π/2, π/2), if y(x) = ∫ (cosec x + sin x) / (cosec x sec x + tan x sin2 x) dx and lim x→π/2 y(x) = 0 then y(π/4) is equal to

Let I(x) = ∫√(x+7)/x dx and I(9) = 12 + 7 loge 7. If I(1) = α + 7 loge (1+2√2), then α^4 is equal to________.

9. $\int e^x \left( \frac{x^2 - 3}{(x - 1)^2} \right) dx$ is equal to -(where C is constant of integration)

∫ (x^2 - 1) / (x^3 * sqrt(2x^4 - 2x^2 + 1)) dx is equal to -

If ∫ \frac{4e^x + 6e^{-x}}{9e^x - 4e^{-x}} dx = Ax + B \ln |9e^{2x} - 4| + C, then

Let I = \int \frac{e^x}{e^{4x} + e^{2x} + 1} dx, J = \int \frac{e^{-x}}{e^{-4x} + e^{-2x} + 1} dx. Then, for an arbitrary constant c, the value of J - I equals

The value of \ \int \cos^{\frac{1}{2}} x \cdot \sin^3 x \, dx\ is _______.

The evaluation of ∫ p xp+2q-1 - q xq-1 ⁄ x2p+2q + 2xp+q + 1 dx is

If \(\int f(\tan x) dx = \lambda\), then \(\int_0^{\pi} f(\tan x) dx\) is equal to

[JEE Main 2021] \(\displaystyle\int x^5\sqrt{1+x^3}\,dx\) equals (where \(C\) is a constant)

\(\int \frac{e^{\tan^{-1} x}}{1+x^2}\,dx\) is equal to

Let f(x) = \frac{2\sin^2 x - 1}{\cos x} + \frac{\cos x(2\sin x + 1)}{1 + \sin x} then \int e^x(f(x) + f'(x))dx (where c is the constant of integration)

Evaluate: \int \frac{\cot x}{(\cos x)^{2005}} dx

The integral \int \frac{(2x - 1) \cos\sqrt{(2x-1)^2 + 5}}{4x^2 - 4x + 6} dx is equal to (where C is a constant of integration) (JEE Main 2021)

If f and g are continuous functions in [0,1] satisfying f(x) = f(a - x) and g(x) + g(a - x) = a, then ∫₀ᵃ f(x) × g(x) dx is equal to

The integral \int \frac{3x^{13} + 2x^{11}}{x^2}dx is equal to (where C is a constant of integration)

The integral \int (x\sin x + \cos x)dx is equal to

The integral ∫√cot x e^√sin x √cos x dx equals

If \int_0^{\pi/3} \frac{\tan\theta}{2k\sec\theta} d\theta = 1 - \frac{1}{\sqrt{2}}, (k > 0), then the value of k is

Evaluate \(\int_{-3}^{3}\sqrt{x^2-1}\,dx\)

$\int \left[ \sin \alpha \sin(x - \alpha) + \sin^2 \left( \frac{x}{2} - \alpha \right) \right] dx$ equals

Evaluate \(\int_{0}^{\pi}\frac{\cos 2x}{\cos^2 x + 1}\,dx\)

Evaluate $\int \frac{dx}{(x+3)^{15/16}(x-4)^{17/16}}$

The integral ∫ sec²x / (sec x + tan x)^9/2 dx equals (for some arbitrary constant K)

\(\displaystyle\int \frac{1}{x^2\sqrt{1-x^2}}\,dx\) equals

If f(x) = ∫x0 g(t) dt, where g is a non-zero even function. If f(x + 5) = g(x), then ∫00 f(t) dt equals

The value of ∫π/2-π/2 \frac{1 + \sin 2x}{1 + \sin^2 x} dx is

If \(\int (e^{2x} + 2e^x \cos x - e^{-x}(e^x + e^{-x})) dx = g(x)e^{(e^x + e^{-x})} + c\), where \(c\) is a constant of integration, then \(g(0)\) is equal to

If the integral \(\int \frac{5\tan x}{\tan x - 2} dx = x + a\log|\sin 2x\cos x| + k\), then \(a\) is equal to

\(\int \frac{2^x + 3^x}{5^x}\) dx is equal to

Let \(f(x) = \int \frac{x^2 \, dx}{(1+x^2)(1+\sqrt{1+x^2})}\) and \(f(0) = 0\), then the value of \(f(1)\) is

\(\int x^x(1 + \log|x|) \, dx\) is equal to

\(\int \frac{x^2}{x^4 + x^2 + 1} dx\) is equal to

If f(x) = ∫₀^x (sin⁴ t + cos⁴ t) dt, then f(x + π) will be equal to:

85. If \(\int f(x) dx = F(x)\), then \(\int x^3 f(x^2) dx\) is equal to

The value of the integral $\int_{0}^{\pi} e^{\cos 2x} \cos^3(2n+1)x dx$ for any integer $n$ has the value

Evaluate \(\int \frac{1}{\cos x} \cdot \sin 2x \cdot \cos 2x \, dx\)

The value of $\displaystyle \int \frac{dx}{x(1+xe^x)^2}$ is equal to

Suppose V = \int_{0}^{\pi/2} \frac{x \sin 2x}{2\sin^2 x + 1} dx, find the value of 96V.

If f(y) = eʸ, g(y) = y, y > 0 and F(t) = ∫₀¹ f(t − y)g(y)dy, then F(t) is