Integrate $w.r.t x, (\log x{)}^2$

Important Questions on Definite Integrals

EASY

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

Let $P\left(x\right)=x^2+bx+c$ be a quadratic polynomial with real coefficients such that ${\int }_0^{1}P\left(x\right)dx=1$ and $P\left(x\right)$ leaves remainder $5$ when it is divided by $\left(x-2\right)$. Then the value of $9\left(b+c\right)$ is equal to:

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Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

${\int }_1^{\sqrt{3}}\frac{dx}{1+x^2}$ equals

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Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

${\int }_1^e\log xdx=$

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Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

If $f:R\to R$ is given by $f\left(x\right)=x+1,$ then the value of $\underset{n\to \infty }{\lim }\frac{1}{n}\left[f\left(0\right)+f\left(\frac{5}{n}\right)+f\left(\frac{10}{n}\right)+\dots ..+f\left(\frac{5(n-1)}{n}\right)\right]$ is:

MEDIUM

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

If $\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\frac{\mathrm{c}\mathrm{o}\mathrm{t}x}{\mathrm{c}\mathrm{o}\mathrm{t}x+\mathrm{cosec}x}dx=m(\mathrm{\pi }+n),$ then $mn$ is equal to

EASY

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

Let $f\left(x\right)=a{x}^2+bx+c,a,b,c\in R,a>0$ and roots of $f\left(x\right)=0$ are not real numbers. The value of ${\int }_0^e\arg \left(-if\left(x\right)\right)dx$ is

EASY

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

The value of ${\int }_0^{\frac{\pi }{2}}\frac{1+2\cos x}{(2+\cos x{)}^2}dx$ is

EASY

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}\left({\sec }^{2}x\right) dx$ is equal to 

MEDIUM

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

The integral ${\int }_{\frac{\mathrm{\pi }}{6}}^{\frac{\mathrm{\pi }}{3}}{\mathrm{s}\mathrm{e}\mathrm{c}}^{\frac{2}{3}}x·\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}{\mathrm{c}}^{\frac{4}{3}}xdx$ is equal to

EASY

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

$\underset{0}{\overset{1}{\int }}x{\left(1-x\right)}^{5}dx=\dots \dots .$

EASY

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

The value of $\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{j=1}^n\frac{\left(2j-1\right)+8n}{\left(2j-1\right)+4n}$ is equal to:

EASY

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

The integral $\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}\frac{dx}{1+\cos x}$ is equal to

EASY

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

${\int }_0^{\frac{\mathrm{\pi }}{2}}\left({\cos }^2\frac{\mathrm{x}}{2}-{\sin }^2\frac{\mathrm{x}}{2}\right)\mathrm{dx}=$

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Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

${\int }_0^{\sqrt{2}}\sqrt{2-x^2}dx=$

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Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

The limit $\underset{n\to \infty }{\lim }\frac{1}{n^{2020}}\sum _{k=1}^n{k}^{2019}$

EASY

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

Find ${\int }_0^{2}f\left(x\right)dx$ ,where $f\left(x\right)=\max \left\{x,x^2\right\}$

EASY

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

If ${\int }_0^k 4x^{3}dx=16$, then the value of $k$ is _____

EASY

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

Let $f$ be a continuous function in $[0,1]$ , then $\underset{n\to \infty }{\lim }\sum _{ j=0}^n\frac{1}{n}f\left(\frac{j}{n}\right)$ is

EASY

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

The integral $\underset{\frac{\pi }{12}}{\overset{\frac{\pi }{4}}{\int }}\frac{8\cos 2x}{{\left(\tan x+\cot x\right)}^3}dx$ equals

MEDIUM

Mathematics>Integral Calculus>Definite Integrals>Evaluation of Definite Integrals

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