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# formulas - Comparison of Diﬀerentiation and integration...

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Unformatted text preview: Comparison of Diﬀerentiation and integration Rules. Let u be a diﬀerentiable function of x. 1. d [x] = 1 dx dn [x ] = nxn−1 , dx 1 d [ln |x|] = , dx x dx [e ] = ex , dx dx [a ] = (ln a)ax , dx d [sin x] = cos x, dx d [cos x] = − sin x, dx d [tan x] = sec2 x, dx d [cot x] = − csc2 x, dx d [sec x] = sec x tan x, dx d [csc x] = − csc x cot x, dx d 1 [arcsin x] = √ , dx 1 − x2 d 1 [arctan x] = , dx 1 + x2 d 1 [arcsec x] = √ , dx x x2 − 1 tan x du = − ln | cos x| + C, cot x du = ln | sin x| + C, sec x du = ln | sec x + tan x| + C, csc x du = − ln | csc x + cot x| + C dx = 1 dx = x + C xn dx = xn+1 +C n+1 du = u u dx = ln |u| + C u 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. dn [u ] = nun−1 u , dx d u [ln |u|] = , dx u du [e ] = eu u , dx du [a ] = (ln a)au u , dx d [sin u] = (cos u) u , dx d [cos u] = −(sin u) u , dx d [tan u] = (sec2 u) u , dx 1 dx = ln |x| + C, x ex dx = ex + C ax dx = 1 ax + C ln a cos x dx = sin x + C sin x dx = − cos x + C sec2 x dx = tan x + C csc2 x dx = − cot x + C sec x tan x dx = secx + C csc x cot x dx = − csc x + C √ x dx = arcsin + C a a2 − x2 d [cot u] = −(csc2 u) u , dx d [sec u] = (sec u tan u) u , dx d [csc u] = −(csc u cot u) u , dx d u [arcsin u] = √ , dx 1 − u2 d u [arctan u] = , dx 1 + u2 d u [arcsec u] = √ , dx u u2 − 1 19. 20. 21. a2 13. 14. 15. 16. 17. 18. dx 1 x = arctan + C 2 +x a a dx 1 x √ = arcsec + C a a x x2 − a2 eax dx = 1 ax e +C a 1 sin ax dx = − cos ax + C a cos ax dx = 1 sin ax + C a 1 ...
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