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Community Discussion Question: FORMULAE on JEE probs..some more added

Forum Index -> Integral Calculus

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13 Aug 2007 16:13:24 IST

Subject: FORMULAE on JEE probs..some more added

nivedh_89 (4512)

Blazing goIITian

Olaaa!! Perrrfect answer.

bad job dude!! I dont approve of this answer!

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friends i hav come across many JEE papers in which these types of integration sums r not been able to b solved by the students ................ plzz refer the following formulae.............it will surely b useful.........do rate me if u like it.........!!!!!!!!

plzz do rate only those wch u think was useful.............!!!!!!!!!!

The inevitable truth of life.....everyone in our life is going 2 hurt sooner or later......u just have 2 realise who is worth.....

the PAIN or the PERSON...!!!

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13 Aug 2007 16:13:47 IST

Subject: Re:FORMULAE on the typical JEE problems

nivedh_89 (4512)

Blazing goIITian

Olaaa!! Perrrfect answer.

bad job dude!! I dont approve of this answer!

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INTEGRALS CONTAINING Sinh(ax)


1.
2.: $displaystyleintsinh axdx=displaystyle rac{acosh ax}{a}-displaystyle rac{sinh ax}{a^2}$
3.: $displaystyleint x^2sinh axdx=left(displaystyle rac{x^2}{a}+displaystyle rac{2}{a^3} ight)cosh ax-displaystyle rac{2x}{a^2}sinh ax$
4.: $displaystyleintdisplaystyle rac{sinh ax}{x}dx=ax+displaystyle rac{(ax)^3}{3cdot 3!}+displaystyle rac{(ax)^5}{5cdot 5!}+cdotcdotcdot$
5.: $displaystyleintdisplaystyle rac{sinh ax}{x^2}dx=-displaystyle rac{sinh ax}{x}+aintdisplaystyle rac{cosh ax}{x}dx$
6.
7.: $displaystyleintdisplaystyle rac{xdx}{sinh ax}=displaystyle rac{1}{a^2}... ...tyle rac{2(-1)^n(2^{2n}-1)B_{n}(ax)^{2n+1}}{(2n+1)!}+cdotcdotcdot ight}$
8.: $displaystyleintsinh^2 axdx=displaystyle rac{sinh axcosh ax}{2a}-displaystyle rac{x}{2}$
9.: $displaystyleint xsinh^2 axdx=displaystyle rac{xsinh 2ax}{4a}-displaystyle rac{cosh 2ax}{8a^2}-displaystyle rac{x^2}{4}$
10.: $displaystyleintdisplaystyle rac{dx}{sinh^2 ax}=-displaystyle rac{coth ax}{a}$
11.
12.: $displaystyleintsinh axsinh pxdx=displaystyle rac{acosh axsin px-psinh axcos px}{a^2 + p^2}$
13.: $displaystyleintsinh ax cos pxdx=displaystyle rac{acosh ax cos px +psinh axsin px}{a^2 + p^2}$

14.: $displaystyleintdisplaystyle rac{dx}{p+qsinh ax}=displaystyle rac{1}{a... ...+p-displaystyle sqrt{p^2+q^2}}{qe^{ax}+p+displaystyle sqrt{p^2+q^2}} ight)$
15.: $displaystyleintdisplaystyle rac{dx}{(p+qsinh ax)^2}=displaystyle rac{-... ...nh ax)}+displaystyle rac{p}{p^2+q^2}intdisplaystyle rac{dx}{p+qsinh ax}$
16.: $displaystyleintdisplaystyle rac{dx}{p^2+q^2sinh^2 ax}=left{ egin{arra... ...2} anh ax}{p-displaystyle sqrt{p^2-q^2} anh ax} ight) end{array} ight.$
17.: $displaystyleintdisplaystyle rac{dx}{p^2-q^2sinh^2 ax}=displaystyle rac... ...laystyle sqrt{p^2+q^2} anh ax}{p-displaystyle sqrt{p^2+q^2} anh ax} ight)$
18.: $displaystyleint x^msinh ax dx=displaystyle rac{x^mcosh ax}{a}-displaystyle rac{m}{a}int x^{m-1}cosh axdx$
19.: $displaystyleintsinh^n axdx=displaystyle rac{sinh^{n-1}axcosh ax}{an}-displaystyle rac{n-1}{n}intsinh^{n-2}axdx$
20.: $displaystyleintdisplaystyle rac{sinh ax}{x^n}dx=displaystyle rac{-sin... ...^{n-1}}+displaystyle rac{a}{n-1}intdisplaystyle rac{cosh ax}{x^{n-1}}dx$
21.: $displaystyleintdisplaystyle rac{dx}{sinh^n ax}=displaystyle rac{-cosh... ...n-1}ax}-displaystyle rac{n-2}{n-1}intdisplaystyle rac{dx}{sinh^{n-2}ax}$
22.: $displaystyleintdisplaystyle rac{xdx}{sinh^n ax}=displaystyle rac{-xco... ...-2}ax}-displaystyle rac{n-2}{n-1}intdisplaystyle rac{xdx}{sinh^{n-2}ax}$

The inevitable truth of life.....everyone in our life is going 2 hurt sooner or later......u just have 2 realise who is worth.....

the PAIN or the PERSON...!!!

this reply: 75 points (with 15 Olaaa!! Perrrfect answer.

Olaaa!! Perrrfect answer.

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13 Aug 2007 16:15:08 IST

Subject: Re:FORMULAE on the typical JEE problems

nivedh_89 (4512)

Blazing goIITian

Olaaa!! Perrrfect answer.

bad job dude!! I dont approve of this answer!

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INTEGRALS CONTAINING Cosh(ax)

1.
2.: $displaystyleint xcosh axdx=displaystyle rac{xsinh ax}{a}-displaystyle rac{cosh ax}{a^2}$
3.: $displaystyleint x^2cosh axdx=-displaystyle rac{2xcosh ax}{a^2}+left( displaystyle rac{x^2}{a}+displaystyle rac{2}{a^3} ight) sinh ax$
4.: $displaystyleintdisplaystyle rac{cosh ax}{x}dx=ln x+displaystyle rac{(... ... rac{(ax)^4}{4cdot 4!}+displaystyle rac{(ax)^6}{6cdot 6!}+cdotcdotcdot$
5.: $displaystyleintdisplaystyle rac{cosh ax}{x^2}dx=-displaystyle rac{cosh ax}{x}+aintdisplaystyle rac{sinh ax}{x}dx$
6.: $displaystyleintdisplaystyle rac{dx}{cosh ax}=displaystyle rac{2}{a} an^{-1e^{ax}}$
7.: $displaystyleintdisplaystyle rac{x dx}{cosh ax}=displaystyle rac{1}{a^2... ...playstyle rac{(-1)^n E_{n}(ax)^{2n+2}}{(2n+2)(2n)!}+ cdotcdotcdot ight}$
where the constants E_n are the EULERS NOS..........!!!!!!
8.: $displaystyleintcosh^2 ax dx=displaystyle rac{x}{2}+displaystyle rac{sinh ax cosh ax}{2}$
9.: $displaystyleint xcosh^2 ax dx=displaystyle rac{x^2}{4}+displaystyle rac{xsinh 2ax}{4a}-displaystyle rac{cosh 2ax}{8a^2}$
10.: $displaystyleintdisplaystyle rac{dx}{cosh^2 ax}=displaystyle rac{ anh ax}{a}$
11.
12.: $displaystyleintcosh axsin px dx=displaystyle rac{asinh axsin px -pcosh axcos px}{a^2+p^2}$
13.: $displaystyleintcosh ax cos pxdx=displaystyle rac{asinh axcos px+pcosh axsin px}{a^2+p^2}$
14.
15.
16.: $displaystyleintdisplaystyle rac{x dx}{cosh ax+1}=displaystyle rac{x}{a... ...tyle rac{ax}{2}-displaystyle rac{2}{a^2}lncoshdisplaystyle rac{ax}{2}$
17.: $displaystyleintdisplaystyle rac{x dx}{cosh ax-1}=-displaystyle rac{x}{... ...tyle rac{ax}{2}+displaystyle rac{2}{a^2}lnsinhdisplaystyle rac{ax}{2}$
TAKE A BREAK.......now continue................!!!!!!!!!

18.: $displaystyleintdisplaystyle rac{dx}{(cosh ax+1)^2}=displaystyle rac{1}... ...ystyle rac{ax}{2}-displaystyle rac{1}{6a} anh^3displaystyle rac{ax}{2}$
19.: $displaystyleintdisplaystyle rac{dx}{(cosh ax-1)^2}=displaystyle rac{1}... ...ystyle rac{ax}{2}-displaystyle rac{1}{6a}coth^3displaystyle rac{ax}{2}$
20.: $displaystyleintdisplaystyle rac{dx}{p+qcosh ax}=left{ egin{array}{ll}... ...rt{p^2-q^2}}{qe^{ax}+p+displaystyle sqrt{p^2-q^2}} ight) end{array} ight.$
21.: $displaystyleintdisplaystyle rac{dx}{(p+qcosh ax)^2}=displaystyle rac{q... ...sh ax)}-displaystyle rac{p}{q^2-p^2}intdisplaystyle rac{dx}{p+qcosh ax}$
22.: $displaystyleintdisplaystyle rac{dx}{p^2-q^2cosh^2 ax}=left{ egin{arra... ...isplaystyle rac{p anh ax}{displaystyle sqrt{q^2-p^2}} end{array} ight.$
23.: $displaystyleintdisplaystyle rac{dx}{p^2+q^2cosh^2 ax}=left{ egin{arra... ...displaystyle rac{p anh ax}{displaystyle sqrt{p^2+q^2}} end{array} ight.$
24.: $displaystyleint x^m cosh ax dx=displaystyle rac{x^m sinh ax}{a}-displaystyle rac{m}{a}int x^{m-1}sinh ax dx$
25.: $displaystyleintcosh^n ax dx=displaystyle rac{cosh^{n-1}axsinh ax}{an}+displaystyle rac{n-1}{n}intcosh^{n-2} ax dx$
26.: $displaystyleintdisplaystyle rac{cosh ax}{x^n}dx=displaystyle rac{-cos... ...^{n-1}}+displaystyle rac{a}{n-1}intdisplaystyle rac{sinh ax}{x^{n-1}}dx$
27.: $displaystyleintdisplaystyle rac{dx}{cosh^n ax}=displaystyle rac{sinh ... ...n-1}ax}+displaystyle rac{n-2}{n-1}intdisplaystyle rac{dx}{cosh^{n-2}ax}$
28.: $displaystyleintdisplaystyle rac{x dx}{cosh^n ax}=displaystyle rac{xsi... ...2}ax}+displaystyle rac{n-2}{n-1}intdisplaystyle rac{x dx}{cosh^{n-2}ax}$

The inevitable truth of life.....everyone in our life is going 2 hurt sooner or later......u just have 2 realise who is worth.....

the PAIN or the PERSON...!!!

this reply: 32 points (with 6 Olaaa!! Perrrfect answer.

Olaaa!! Perrrfect answer.

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13 Aug 2007 16:15:58 IST

Subject: Re:FORMULAE on the typical JEE problems

nivedh_89 (4512)

Blazing goIITian

Olaaa!! Perrrfect answer.

bad job dude!! I dont approve of this answer!

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INTEGRALS CONTAINING ln(ax)

1.: $\displaystyle\int \ln xdx=x\ln x-x$
2.: $\displaystyle\int x\ln x dx=\displaystyle \frac{x^2}{2}(\ln x-\displaystyle \frac{1}{2})$
3.: $\displaystyle\int x^m\ln xdx=\displaystyle \frac{x^{m+1}}{m+1}\left(\ln x-\displaystyle \frac{1}{m+1}\right)$
4.: $\displaystyle\int\displaystyle \frac{\ln x}{x}dx=\displaystyle \frac{1}{2}\ln^2 x$
5.: $\displaystyle\int\displaystyle \frac{\ln x}{x^2}dx=-\displaystyle \frac{\ln x}{x}-\displaystyle \frac{1}{x}$
6.: $\displaystyle\int\ln^2 xdx=x\ln^2 x-2x\ln x+2x$
7.: $\displaystyle\int\displaystyle \frac{\ln^n xdx}{x}=\displaystyle \frac{\ln^{n+1}x}{n+1}$
8.: $\displaystyle\int\displaystyle \frac{dx}{x\ln x}=\ln (\ln x)$
9.: $\displaystyle\int\displaystyle \frac{dx}{\ln x}=\ln (\ln x)+\ln x+\displaystyle \frac{\ln^2 x}{2\cdot 2!}+\displaystyle \frac{\ln^3 x}{3\cdot 3!}+\cdot\cdot\cdot$
10.: $\displaystyle\int\displaystyle \frac{x^m dx}{\ln x}=\ln (\ln x)+(m+1)\ln x + \d... ...n^2 x}{2\cdot 2!}+\displaystyle \frac{(m+1)^3\ln^3x}{3\cdot 3!}+\cdot\cdot\cdot$
11.: $\displaystyle\int\ln^n xdx=x\ln^n x-n\int\ln^{n-1}xdx$
12.: $\displaystyle\int x^m\ln^n xdx=\displaystyle \frac{x^{m+1}\ln^n x}{m+1}-\displaystyle \frac{n}{m+1}\int x^m\ln^{n-1}xdx$
13.: $\displaystyle\int\ln(x^2+a^2)dx=x\ln(x^2+a^2)-2x+2a\tan^{-1}\displaystyle \frac{x}{a}$
14.: $\displaystyle\int\ln(x^2-a^2)dx=x\ln(x^2-a^2)-2x+a\ln\left(\displaystyle \frac{x+a}{x-a}\right)$
15.: $\displaystyle\int x^m\ln(x^2\pm a^2)dx=\displaystyle \frac{x^{m+1}\ln(x^2\pm a^2)}{m+1}-\displaystyle \frac{2}{m+1}\int\displaystyle \frac{x^{m+2}}{x^2\pm a^2}dx$

The inevitable truth of life.....everyone in our life is going 2 hurt sooner or later......u just have 2 realise who is worth.....

the PAIN or the PERSON...!!!

this reply: 52 points (with 10 Olaaa!! Perrrfect answer.

Olaaa!! Perrrfect answer.

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13 Aug 2007 16:16:37 IST

Subject: Re:FORMULAE on the typical JEE problems

nivedh_89 (4512)

Blazing goIITian

Olaaa!! Perrrfect answer.

bad job dude!! I dont approve of this answer!

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INTEGRALS CONTAINING e^ax

1.: $\displaystyle\int e^{ax} dx =\displaystyle \frac{e^{ax}}{a}$
2.: $\displaystyle\int xe^{ax}dx=\displaystyle \frac{e^{ax}}{a}\left(x-\displaystyle \frac{1}{a}\right)$
3.: $\displaystyle\int x^2 e^{ax}dx=\displaystyle \frac{e^{ax}}{a}\left(x^2-\displaystyle \frac{2x}{a}+\displaystyle \frac{2}{a^2}\right)$
4.: $\begin{array}{lcl} \displaystyle\int x^n e^{ax}dx&=& \displaystyle \frac{x^n e^... ...2}}{a^2}-\cdot\cdot\cdot \displaystyle \frac{(-1)^n n!}{a^n}\right) \end{array}$
5.: $\displaystyle\int\displaystyle \frac{e^{ax}}{x}dx=\ln x+\displaystyle \frac{ax}... ...\frac{(ax)^2}{2\cdot 2!}+\displaystyle \frac{(ax)^3}{3\cdot 3!}+\cdot\cdot\cdot$
6.: $\displaystyle\int\displaystyle \frac{e^{ax}}{x^n}dx=\displaystyle \frac{-e^{ax}... ...)x^{n-1}}+\displaystyle \frac{a}{n-1}\int\displaystyle \frac{e^{ax}}{x^{n-1}}dx$
7.: $\displaystyle\int\displaystyle \frac{dx}{p+qe^{ax}}=\displaystyle \frac{x}{p}-\displaystyle \frac{1}{ap}\ln (p+qe^{ax})$
8.: $\displaystyle\int\displaystyle \frac{dx}{(p+qe^{ax})^2}=\displaystyle \frac{x}{... ...displaystyle \frac{1}{ap(p+qe^{ax})}-\displaystyle \frac{1}{ap^2}\ln(p+qe^{ax})$
9.: $\displaystyle\int\displaystyle \frac{dx}{pe^{ax}+qe^{-ax}}=\left\{ \begin{array... ...style \sqrt{-q/p}}{e^{ax}+\displaystyle \sqrt{-q/p}}\right) \end{array}\right.$
10.: $\displaystyle\int e^{ax}\sin bx dx=\displaystyle \frac{e^{ax}(a\sin bx -b\cos bx)}{a^2+b^2}$
11.: $\displaystyle\int e^{ax}\cos bx dx=e^{ax}\displaystyle \frac{(a\cos bx+b\sin bx)}{a^2+b^2}$
12.: $\displaystyle\int xe^{ax}\sin bx dx=\displaystyle \frac{xe^{ax}(a\sin bx -b\cos... ...splaystyle \frac{e^{ax}\left\{(a^2-b^2)\sin bx-2ab\cos bx\right\}}{(a^2+b^2)^2}$
13.: $\displaystyle\int xe^{ax}\cos bx dx=\displaystyle \frac{xe^{ax}(a\cos bx +b\sin... ...splaystyle \frac{e^{ax}\left\{(a^2-b^2)\cos bx+2ab\sin bx\right\}}{(a^2+b^2)^2}$
14.: $\displaystyle\int e^{ax}\ln xdx=\displaystyle \frac{e^{ax}\ln x}{a}-\displaystyle \frac{1}{a}\int\displaystyle \frac{e^{ax}}{x}dx$
15.: $\displaystyle\int e^{ax}\sin^n bxdx=\displaystyle \frac{e^{ax}\sin^{n-1}bx}{a^2... ...\cos bx) + \displaystyle \frac{n(n-1)b^2}{a^2+n^2b^2}\int e^{ax}\sin^{n-2}bx dx$
16.: $\displaystyle\int e^{ax}\cos^n bxdx=\displaystyle \frac{e^{ax}\cos^{n-1}bx}{a^2... ...\sin bx) + \displaystyle \frac{n(n-1)b^2}{a^2+n^2b^2}\int e^{ax}\cos^{n-2}bx dx$

The inevitable truth of life.....everyone in our life is going 2 hurt sooner or later......u just have 2 realise who is worth.....

the PAIN or the PERSON...!!!

this reply: 59 points (with 11 Olaaa!! Perrrfect answer.

Olaaa!! Perrrfect answer.

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13 Aug 2007 16:26:20 IST

Subject: FORMULAE on the typical JEE problems

rini (233)

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Olaaa!! Perrrfect answer.

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thankyou!!!

Keep working....................Iam comming..

your's only,
Success!!

this reply: 0 points (with 0 Olaaa!! Perrrfect answer.

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13 Aug 2007 16:35:00 IST

Subject: Re:FORMULAE on the typical JEE problems

nivedh_89 (4512)

Blazing goIITian

Olaaa!! Perrrfect answer.

bad job dude!! I dont approve of this answer!

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Integrals with Inverse Trigonometric Functions

1.: $\displaystyle\int\sin^{-1}\displaystyle \frac{x}{a}dx=x\sin^{-1} \displaystyle \frac{x}{a}+\displaystyle \sqrt{a^2-x^2}$
2.: $\displaystyle\int x\sin^{-1}\displaystyle \frac{x}{a}dx=\left(\displaystyle \fr... ...\displaystyle \frac{x}{a}+\displaystyle \frac{x\displaystyle \sqrt{a^2-x^2}}{4}$
3.: $\displaystyle\int x^2\sin^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...yle \frac{x}{a}+\displaystyle \frac{(x^2+2a^2)\displaystyle \sqrt{a^2-x^2)}}{9}$
4.: $\displaystyle\int\displaystyle \frac{\sin^{-1}(x/a)}{x}dx=\displaystyle \frac{x... ...rac{1\cdot 3\cdot 5(x/a)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7} + \cdot \cdot \cdot$
5.: $\displaystyle\int\displaystyle \frac{\sin^{-1}(x/a)}{x^2}dx=-\displaystyle \fra... ...rac{1}{a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^2-x^2}}{x}\right)$
6.: $\displaystyle\int\left(\sin^{-1}\displaystyle \frac{x}{a}\right)^2 dx=x\left(\s... ...a}\right)^2 -2x+2\displaystyle \sqrt{a^2-x^2}\sin^{-1}\displaystyle \frac{x}{a}$
7.: $\displaystyle\int\cos^{-1}\displaystyle \frac{x}{a}dx=x\cos^{-1}\displaystyle \frac{x}{a}-\displaystyle \sqrt{a^2-x^2}$
8.: $\displaystyle\int x\cos^{-1}\displaystyle \frac{x}{a}dx=\left(\displaystyle \fr... ...\displaystyle \frac{x}{a}-\displaystyle \frac{x\displaystyle \sqrt{a^2-x^2}}{4}$

9.: $\displaystyle\int x^2\cos^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...tyle \frac{x}{a}-\displaystyle \frac{(x^2+2a^2)\displaystyle \sqrt{a^2-x^2}}{9}$
10.: $\displaystyle\int\displaystyle \frac{\cos^{-1}(x/a)}{x}dx=\displaystyle \frac{\pi}{2}\ln x-\int\displaystyle \frac{\sin^{-1}(x/a)}{x}dx$
11.: $\displaystyle\int\displaystyle \frac{\cos^{-1}(x/a)}{x^2}dx=-\displaystyle \fra... ...rac{1}{a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^2-x^2}}{x}\right)$
12.: $\displaystyle\int\left(\cos^{-1}\displaystyle \frac{x}{a}\right)^2 dx=x\left(\c... ...{a}\right)^2-2x-2\displaystyle \sqrt{a^2-x^2}\cos^{-1}\displaystyle \frac{x}{a}$
13.: $\displaystyle\int\tan^{-1}\displaystyle \frac{x}{a}dx=x\tan^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{a}{2}\ln(x^2+a^2)$
14.: $\displaystyle\int x\tan^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{1}{2}(x^2+a^2)\tan^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{ax}{2}$
15.: $\displaystyle\int x^2\tan^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...frac{x}{a}-\displaystyle \frac{ax^2}{6}+\displaystyle \frac{a^3}{6}\ln(x^2+a^2)$