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Integrals of Exponential and Logarithmic Functions. ∫ ln x dx = x ln x − x + C. + 1 x. + 1. x ∫ x ln xdx = ln x − + C. 2 + 1 ( n + 1 ) x dx = e x + C ∫.
Page 2 of 6. INTEGRATION FORMULAE STANDARD INTEGRALS 1. ∫, ( )- ( ) , ( )- [where, n ≠ 1- 2. ∫ ( ) ( ) * ( )+ 3. ∫ ( ) ∫ ( ) [where, v be the function of x]
Integration by parts: u dv = uv. −. v du + C. ax + b. Partial Fractions: to integrate a function like : (x + c)(x + d) ax + b A B A(x + d) + B(x + c) Write = + = , (x + c)(x + d) (x + c) (x + d) (x + c)(x + d) so ax + b = A(x + d) + B(x + c) = (A + B)x + (Ad + Bc), so a = A + B and b = Ad + Bc; solve for A and B .
Sometimes we can rewrite an integral to match it to a standard form. More often however, we will need more advanced techniques for solving integrals. First, let’s look at some examples of our known methods. Basic integration formulas. 1. k dx = kx + C. xn+1. 2. xndx = + C. + 1. 3. dx = ln |x| + C. x. 4. ex dx = ex + C. 5. axdx ax. = + C ln(a)
Integration Formulas: xp+1. xp dx = + C; p 6=. + 1. (2) sin(x) dx = cos(x) + C. Z. (3) cos(x) dx = sin(x) + C. Z.
Basic Integration Formulas 1. Power functions: (1) Z xn = xn+1 n+1 +C,n 6= −1 (2) Z 1 x dx = ln|x|+C 2. Trigonometric functions: (3) Z sinxdx = −cosx+C (4) Z cosxdx = sinx+C (5) Z sec2xdx = tanx+C (6) Z csc2 xdx = −cotx+C (7) Z secxtanxdx = secx+C (8) Z cscxcotx = −cscx+C 3. Exponential function: (9) Z exdx = ex +C 4. Inverse of ...
Basic integration formulas. (x + c)(x + d) (x + c) (x + d) (x + c)(x + d) so ax + b = A(x + d) + B(x + c) = (A + B)x + (Ad + Bc), so a = A + B and b = Ad + Bc; solve for A and B . The approach for more general denomenator can be found in nearly any calculus textbook.
