Embibe Experts Solutions for Chapter: Definite Integration, Exercise 1: JEE Main - 1 February 2023 Shift 1

Let $\alpha \in \left(0,1\right)$ and $\beta ={\log }_e\left(1-\alpha \right)$. Let $P_n\left(x\right)=x+\frac{x^2}{2}+\frac{x^3}{3}+\dots .+\frac{x^n}{n},x\in \left(0,1\right)$. Then the integral ${\int }_0^{\alpha }\frac{t^{50}}{1-t}dt$ is equal to

The value of ${\int }_{\frac{\pi }{3}}^{\frac{\pi }{2}}\frac{\left(2+3\sin x\right)}{\sin x\left(1+\cos x\right)}dx$ is equal to

If $\varphi \left(x\right)=\frac{1}{\sqrt{x}}{\int }_{\frac{\pi }{4}}^x\left(4\sqrt{2}\sin t-3{\varphi }^{\text{'}}\left(t\right)\right)dt,x>0$ then ${\varphi }^{\text{'}}\left(\frac{\pi }{4}\right)$ is equal to

Let $\alpha >0$. If ${\int }_0^{\alpha }\frac{x}{\sqrt{x+\alpha }-\sqrt{x}}dx=\frac{16+20\sqrt{2}}{15}$ then $\alpha$ is equal to :

$\underset{n\to \infty }{\lim }\left(\frac{1}{1+n}+\frac{1}{2+n}+\frac{1}{3+n}+\dots +\frac{1}{2n}\right)$  is equal to :-

If
${\int }_0^1\left(x^{21}+x^{14}+x^7\right){\left(2x^{14}+3x^7+6\right)}^{1/7}dx=\frac{1}{l}{\left(11\right)}^{m/n}$ where $l, m,n\in N,m$ and $n$ are co-prime then $l+m+n$ is equal to _____ .

The value of the integral ${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\frac{x+\frac{\pi }{4}}{2-\cos 2x}dx$ is :

If ${\int }_0^{\pi }\frac{5^{\cos x}\left(1+\cos x\cos 3x+{\cos }^{2}x+{\cos }^{3}x\cos 3x\right)dx}{1+5^{\cos x}}=\frac{k\pi }{16}$, then $k$ is equal to _____ .