 Calculus 1: Integrals questions and answers

Recent questions in Integrals oliviayychengwh 2022-01-07 Answered

Compute the following integral: $$\displaystyle{\int_{{0}}^{\infty}}{\frac{{{e}^{{x}}{\sin{{\left({x}\right)}}}}}{{{x}}}}{\left.{d}{x}\right.}$$ abreviatsjw 2022-01-07

How to evaluate the integral $$\displaystyle\int{e}^{{{x}^{{3}}}}{\left.{d}{x}\right.}$$ Michael Maggard 2022-01-07 Answered

I can't find a good way to integrate: $$\displaystyle\int{\frac{{{3}{\sin{{\left({x}\right)}}}+{2}{\cos{{\left({x}\right)}}}}}{{{2}{\sin{{\left({x}\right)}}}+{3}{\cos{{\left({x}\right)}}}}}}{\left.{d}{x}\right.}$$ elvishwitchxyp 2022-01-06 Answered

How to evaluate the following improper integral : $$\displaystyle{\int_{{0}}^{{+\infty}}}{\frac{{{x}{\sin{{x}}}}}{{{x}^{{2}}+{1}}}}{\left.{d}{x}\right.}$$ Michael Maggard 2022-01-06 Answered

Evaluate the following integral: $$\displaystyle\int{\frac{{\sqrt{{{\sin{{x}}}}}}}{{\sqrt{{{\sin{{x}}}}}+\sqrt{{{\cos{{x}}}}}}}}{\left.{d}{x}\right.}$$ David Lewis 2022-01-06 Answered

Evaluate $$\displaystyle{\int_{{0}}^{{1}}}{\left({\frac{{{1}}}{{{\ln{{x}}}}}}+{\frac{{{1}}}{{{1}-{x}}}}\right)}^{{2}}{\left.{d}{x}\right.}$$ William Cleghorn 2022-01-06 Answered

Prove that: $$\displaystyle{\int_{{0}}^{{1}}}{\frac{{{x}{\ln{{\left({1}+{x}\right)}}}}}{{{1}+{x}^{{2}}}}}{\left.{d}{x}\right.}={\frac{{\pi^{{2}}}}{{{96}}}}+{\frac{{{{\ln}^{{2}}{2}}}}{{{8}}}}$$ kramtus51 2022-01-06 Answered

Consider the following integral: $$\displaystyle{I}={\int_{{0}}^{\infty}}{\frac{{{x}-{1}}}{{\sqrt{{{2}^{{x}}-{1}}}{\ln{{\left({2}^{{x}}-{1}\right)}}}}}}{\left.{d}{x}\right.}$$ Wanda Kane 2022-01-06 Answered

Evaluate integral: $$\displaystyle{\int_{{0}}^{\infty}}{\frac{{{\ln{{x}}}}}{{{1}+{x}^{{2}}}}}{\left.{d}{x}\right.}$$ Gregory Jones 2022-01-05

I have in trouble for evaluating following integral $$\displaystyle{\int_{{0}}^{\infty}}{\left(\sqrt{{{1}+{x}^{{4}}}}-{x}^{{2}}\right)}{\left.{d}{x}\right.}={\frac{{\Gamma^{{2}}{\left({\frac{{{1}}}{{{4}}}}\right)}}}{{{6}\sqrt{{\pi}}}}}$$ piarepm 2022-01-05 Answered

Integrate: $$\displaystyle\int{\frac{{{x}^{{2}}}}{{{\left({x}{\cos{{x}}}-{\sin{{x}}}\right)}{\left({x}{\sin{{x}}}+{\cos{{x}}}\right)}}}}{\left.{d}{x}\right.}$$ James Dale 2022-01-05 Answered

Prove: $$\displaystyle{\int_{{0}}^{{\infty}}}{x}^{{{2}{n}}}{e}^{{-{x}^{{2}}}}{\left.{d}{x}\right.}={\frac{{{\left({2}{n}\right)}!}}{{{2}^{{{2}{n}}}{n}!}}}{\frac{{\sqrt{{\pi}}}}{{{2}}}}$$ Mary Reyes 2022-01-05 Answered

How to compute $$\displaystyle{\int_{{-\infty}}^{\infty}}{\exp{{\left(-{\frac{{{\left({x}^{{2}}-{13}{x}-{1}\right)}^{{2}}}}{{{611}{x}^{{2}}}}}\right)}}}{\left.{d}{x}\right.}$$ Adela Brown 2022-01-05 Answered

Evaluate $$\displaystyle\int{\frac{{{1}}}{{{\left({x}^{{2}}+{1}\right)}^{{2}}}}}{\left.{d}{x}\right.}$$ Adela Brown 2022-01-05 Answered

Evaluate $$\displaystyle{\int_{{0}}^{{{2}\pi}}}{\frac{{{1}}}{{{\left({1}+{a}{\cos{\theta}}\right)}^{{2}}}}}{d}\theta,\ {0}\leq{a}{ < }{1}$$ Algotssleeddynf 2022-01-05 Answered

I cannot prove that if $$\displaystyle{f{{\left({x}\right)}}}$$ is convex on $$\displaystyle{\left[{a},{b}\right]}$$ then $$\displaystyle{f{{\left({\frac{{{a}+{b}}}{{{2}}}}\right)}}}\leq{\frac{{{1}}}{{{b}-{a}}}}{\int_{{a}}^{{b}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\leq{\frac{{{f{{\left({a}\right)}}}+{f{{\left({b}\right)}}}}}{{{2}}}}$$ Miguel Reynolds 2022-01-05 Answered

What is the proof of the following: $$\displaystyle{\int_{{0}}^{{1}}}{\left({\frac{{{\ln{{t}}}}}{{{1}-{t}}}}\right)}^{{2}}{\left.{d}{t}\right.}={\frac{{\pi^{{2}}}}{{{3}}}}$$ ? James Dale 2022-01-05 Answered

Evaluate the integral: $$\displaystyle{\int_{{0}}^{{1}}}{\left({\sqrt[{{3}}]{{{1}-{x}^{{7}}}}}-{\sqrt[{{7}}]{{{1}-{x}^{{3}}}}}\right)}{\left.{d}{x}\right.}$$ Roger Smith 2022-01-05 Answered

Can this integral be solved with contour integral or by some application of residue theorem? $$\displaystyle{\int_{{0}}^{\infty}}{\frac{{{\log{{\left({1}+{x}\right)}}}}}{{{1}+{x}^{{2}}}}}{\left.{d}{x}\right.}={\frac{{\pi}}{{{4}}}}{\log{{2}}}+\ \text{ Catalan constant}$$ Sapewa 2022-01-05 Answered

Evaluate the integral $$\displaystyle{\int_{{-\infty}}^{{\infty}}}{\frac{{{\cos{{\left({x}\right)}}}}}{{{x}^{{2}}+{1}}}}{\left.{d}{x}\right.}$$

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