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In what follows, c is a constant of integration and can take any contant value.
1 - Integrals of Elementary Functions.
1.1 ò dx = x + c
1.2 ò k dx = k x + c , where k is a constant.
1.3 ò x n dx = x n + 1 / (n + 1) + c
1.4 ò (1 / x) dx = ln |x| + c
2 - Integrals of Elementary Trigonometric Functions : sin x, cos x, tan x, cot x, sec x and csc x.
2.1 ò sin x dx = -cos x + c
2.2 ò cos x dx = sin x + c
2.3 ò tan x dx = ln |sec x| + c
2.4 ò cot x dx = ln |sin x| + c
2.5 ò sec x dx = ln |sec x + tan x| + c
2.6 ò csc x dx = ln |csc x - cot x| + c
3 - Integrals Involving More Than One Trigonometric Function.
3.1 ò sec x tan x dx = sec x + c
3.2 ò csc x cot x dx = -csc x + c
3.3 ò sin mx sin nx dx =
-sin [(m + n)x] / 2(m + n) + sin [(m - n)x] / 2(m - n) + c
with m not equal to n.
3.4 ò cos mx cos nx dx =
sin [(m + n)x] / 2(m + n) + sin [(m - n)x] / 2(m - n) + c
with m not equal to n.
3.5 ò sin mx cos nx dx =
-cos [(m + n)x] / 2(m + n) - cos [(m - n)x] / 2(m - n) + c
with m not equal to n.
4 - Integrals Involving Exponential and Logarithmic Functions.
4.1 ò e x dx = e x + c
4.2 ò a x dx = a x / ln a + c
4.3 ò ln x dx = x ln x - x + c
5 - Integrals of Inverse Trigonometric functions: arcsin x, arccos x, arctan x, arccot x, arcsec x and arccsc x.
5.1 ò arcsin x dx = x arcsin x + sqrt (1 - x 2) + c
5.2 ò arccos x dx = x arccos x - sqrt (1 - x 2) + c
5.3 ò arctan x dx = x arctan x - ln [sqrt (1 + x 2)] + c
5.4 ò arccot x dx = x arccot x + ln sqrt (1 + x 2) + c
5.5 ò arcsec x dx = x arcsec x - ln [x + sqrt (x 2 - 1)] + c
5.6 ò arccsc x dx = x arccsc x + ln [x + sqrt (x 2 - 1)] + c
6 - Integrals Involving Exponential and Sine and Cosine Functions.
6.1 òe ax sin bx dx = (e ax / (a 2 + b 2) (a*sin bx - b*cos bx) + c
6.2 òe ax cos bx dx = (e ax / (a 2 + b 2) (b*sin bx + a*cos bx) + c
7 - Integrals Involving Hyperbolic Functions: sinh x, cosh x, tanh x, coth x, sech x, csch x.
7.1 ò sinh x dx = cosh x + c
7.2 ò cosh x dx = sinh x + c
7.3 ò sech x tanh x dx = -sech x + c
7.4 ò csch x coth x dx = -csch x + c
7.5 ò sech 2 x dx = tanh x + c
7.6 ò csch 2 x dx = -coth x + c
More references on
integrals and their applications in calculus.
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