Definite Integration Questions (1340)

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

Suppose that F(x) is an antiderivative of f(x) = (sin x)/(x) − (3 sin 2x)/(1 + x²), where x > 0, then ∫ dx can be

∫₀² (ln x)/(x² + 1 + x²)dx is equal to

Evaluate \(\int_3^5 x^5 \sqrt[3]{1 + 3x^4} \, dx\)

Evaluate I = ∫02π [sin x + cos x] dx, where [·] is the greatest integer function.

Evaluate $\displaystyle \int_0^{\pi/2} \sin x \, dx$

If \(f(x) = A \sin(x/2) + B\), \(f'(\pi/2) = 2\), and \(\sqrt{2}\int_0^1 f(x)dx = \frac{2A}{\pi}\), then the constants A and B are

If \(f : \mathbb{R} \to \mathbb{R}\) is continuous and differentiable function such that \(\int_0^x f(t) dt + \int_0^x t^3 dt = \int_0^x t^2 dt + f(1) \int_0^x f(t) dt - f(2) \int_0^x t dt + f(3)\), then the value of \(f(4)\) is

Let $f(x) = \frac{x}{(1+nx^n)^{1/n}}$ for $n \ge 2$ and $g(x) = \underbrace{(f \circ f \circ \dots \circ f)}_{f \text{ occurs } n \text{ times}}(x)$. Then $\int x^{n-2} g(x) dx$ equals.

The value of \(I = \int_0^{\pi/2} \dfrac{(\sin x + \cos x)^2}{\sqrt{1 + \sin 2x}}\, dx\) is

If \(f\left(\dfrac{x-4}{x+2}\right) = 2x + 1\), \((x \in \mathbb{R} - \{1, -2\})\), then \(\int f(x)\, dx\) is equal to (where \(C\) is a constant of integration):