Definite Integration Questions (1340)

Find $\int \frac{2(1+x^2) + 3\sqrt{1-x^2}}{(1+x^2)\sqrt{1-x^2}} dx$

Let I(x) = ∫ \frac{x^2(\sec^2 x + \tan x)}{(x \tan x + 1)^2} dx. If I(0) = 0 then I\left(\frac{\pi}{4}\right) is equal to

Let $\int_0^x\sqrt{1-(y'(t))^2}\,dt=\int_0^x y(t)\,dt$, $0\leq x\leq3$, $y\geq0$, $y(0)=0$. Then at $x=2$, $y''+y+1$ is equal to:

Evaluate \(\displaystyle\int_0^{\pi/2}\sqrt{1-\sin 2x}\,dx\) [JEE Main 2020]

Evaluate the integral: $$I = \int \frac{\tan\left(\frac{\pi}{4} - x\right)}{\cos^2 x \sqrt{\tan^3 x + \tan^2 x + \tan x}} dx$$

∫ \frac{x^2 + 3}{x^6(x^2 + 1)} dx equals

If $f(x) = \begin{cases} 1-|x|, & |x| \le 1 \\ |x|-1, & |x| > 1 \end{cases}$, and $g(x) = f(x-1) + f(x+1)$. Find the value of $\int_{-3}^{5} g(x) dx$.

Evaluate: $\int \frac{1}{x^4 + 5x^2 + 1} dx$

Find $\int \frac{1 + \sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x} dx$

If f(x) = x/(1 + (ln x)^{(ln x)^{...∞}}) for all x ∈ [1, ∞), then ∫₁^{e²} f(x)dx equals:

The value of \(\lim_{h \to 0} \frac{1}{h} \int_{1}^{1+2h} e^{\sqrt{x}} \sin\left(\frac{\pi x}{3}\right) dx\) equals:

Evaluate $\int \left( \ln(\ln x^2) + \frac{2}{\ln x^2} \right) dx$

Evaluate $\int_{0}^{\pi/4} \sin^5 2x \, dx$.

Evaluate $\int_{0}^{\pi} \cot x \cos 2x \, dx$

Prove that $\int e^{g(x)} (g'(x) \cdot f(x) + f'(x)) dx = e^{g(x)} \cdot f(x)$.

∫ \frac{x^2(1-\ln x)}{\ln^4 x - x^4} dx equals

Estimate the value of $\int_{0}^{1} e^{x^2} dx$ using (i) rectangle, (ii) triangle.

Evaluate $\int \frac{x(1-x^2)}{1+x^4} dx$

Evaluate $\int_{\pi/2}^{\pi} x^{\sin x} (1 + x \cos x \cdot \ln x + \sin x) dx$

Evaluate $\int \sin^3 x \cos^5 x dx$

Evaluate $\int \frac{x + \sin x}{1 + \cos x} dx$

\(\int\limits_0^{\frac{\pi }{2}} {[\sin x + \cos x]} \)dx, where [ ] represents the greatest integer function, equals

$\int \frac{dx}{x(x+1)}$ is equal to (where $C$ is an arbitrary constant)

Evaluate $\int \cos \sqrt{x} \, dx$

Find $\int \frac{5\sin x}{\sin x - 2\cos x} dx$

Evaluate $\int \frac{3x+2}{4x^2+4x+5} dx$

If \(f(x) = \frac{x-1}{x+1}\) and \(g(x) = f^{-1}(x)\), then \(\int_{1/e}^1 g(x) dx\) is equal to

If $H(x) = \sin(x^2)$, whose range is $[-1, 1]$. $a = -1, b = 1 \Rightarrow a + 2b = 1$

$(C) \, 2\tan^{-1}(e^x\sin x) + C$

If $I_{m,n} = \int_{0}^{\pi/2} \sin^m x \cdot \cos^n x dx$, then show that $I_{m,n} = \frac{m-1}{m+n} I_{m-2,n}$

Evaluate \(\int \frac{x+2}{\sqrt{x^2+2x+3}} dx\)

Evaluate \(\int \frac{dx}{9x^2 + 6x + 5}\)

∫ x2 + 3 / x6(x2 + 1) dx equals

If for all real triplets \(a, b, c\), \(f(x) = a + bx + cx^2\), then \(\int_0^1 f(x)\, dx\) is equal to:

If the value of the integral \[\int_0^{1/2} \frac{x}{(1-x^2)^{3/2}} dx\] is \[\frac{k}{6}\], then \(k\) is equal to (JEE Main 2020)

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

If ∫ \(\frac{\cos 4x}{\sin^2 x}\) dx = \(A \cot x + B \sin 2x + C\), then find A and B.

Evaluate $\int x^2 e^x \, dx$

Prove that $\int_{0}^{\pi/2} \log(\sin x) dx = \int_{0}^{\pi/2} \log(\cos x) dx = -\frac{\pi}{2} \log 2$

Let I(x) = ∫ 6 / (sin² x (1 - cot x)²) dx. If I(0) = 3, then I(π/12) is equal to:

\(\int \frac{3\sin x + 2\cos x}{3\cos x + 2\sin x} dx\) is equal to

If $C = \int_0^1 [F(x+a) - F(1+a)]dx$, then $C =$

If $I = 2\int_0^1 x\sin(\pi x)dx$, then $I =$

[JEE Main 2021] \(\displaystyle\int\frac{dx}{x^2(x^4+1)^{3/4}}\) equals (where \(C\) is constant)

86. The anti-derivative of \(f(x) = \log(\log x) + (\log x)^{-2}\) whose graph passes through \((e, e)\), is

If $$I_n = \int_0^{\pi} \frac{\sin(2nx)}{\sin^2 x} \, dx$$, then the value of $$I_{n+1}$$ is equal to (where $$n \in \mathbb{I}$$):

The \(L\) denotes the value of the definite integral \(\displaystyle\int_0^1 \dfrac{1}{1+x^8}\,dx\), then which one of the following must be true?

If \(xf(x) = 3f^2(x) + 2\), then \(\int \frac{2x - 12xf(x) - f(x)}{(6f(x) - x)(x^2 - f(x))^2} dx\) equals

Evaluate \(\int \frac{\sin 2x}{\sin^4 x} dx\)

Let \( I = \int_0^{5\pi/12} [\tan x] \, dx \), where \([\cdot]\) denotes the greatest integer function. Find the value of \(I\).