Definite Integration

$\lim\limits_{n \to \infty} \dfrac{1}{\sqrt{n}}\left[1 + \dfrac{1}{\sqrt{2}} + \dfrac{1}{\sqrt{3}} + \cdots + \dfrac{1}{\sqrt{n}}\right]$

$\displaystyle\int_{\frac{\pi}{2}}^{\pi} \dfrac{\sin x - x\cos x}{x(x + \sin x)}\, dx$ equals

The limit $\lim _{a \rightarrow 0} \frac{1}{a} \int_{2-a}^{2+a} \cos ^2(\pi t) e^{\left|\frac{2-t}{a}\right|} d t$

The value of $\sum_{k=1}^{2025} \int_{2 k \pi}^{2 k \pi+\frac{\pi}{2}} \frac{\cos ^k(x)}{\cos ^k(x)+\sin ^k(x)} d x$ is

The value of the integral $\int_0^{\sqrt{\pi}} x \sin ^2\left(x^2\right) d x$ is

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. For $0 \lt t \lt \pi$, let $R_t$ be the region bounded by the curve $y=f(x)$, the line $x=0$, the line $y=0$ and the line $x=t$. Suppose the area of…

An example of a correct inequality is

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function satisfying $f^{\prime}=f$ and $f(0)=1$. Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be the function that satisfies $f(x)+g(x)=x^3, x \in \mathbb{R}$ Then…

Let $I=\int_0^1 \frac{x^7}{\sqrt[3]{1+x^8}} d x$ Then

Let $a \gt 0$ and $f, g$ be continuous functions on $[0, a]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=3$. Then $\int_0^a f(x) g(x) d x$ is

Let $f:[0,2 \pi] \rightarrow \mathbb{R}$ be a differentiable, strictly increasing function such that $f(0) \lt 0 \lt f(2 \pi)$. Let $F(t)=\int_0^t f(x) d x$ for all $t$ in $[0,2 \pi]$. Then

For $a, b \gt 0$ let $F(a, b)=\int_a^b|\sin 2 \pi x| d x$. Then