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A Review: The basic integration formulas summarise the forms of indefinite integrals for may of the functions we have studied so far, and the substitution method helps us use the table below to evaluate more complicated functions involving these basic ones.
Integration Formulas. 1. Common Integrals. Indefinite Integral. Method of substitution. ∫ f ( g ( x )) g ′ ( x ) dx = ∫ f ( u ) du. Integration by parts. ∫ f ( x ) g ′ ( x ) dx = f ( x ) g ( x ) − ∫ g ( x ) f ′ ( x ) dx. Integrals of Rational and Irrational Functions. + 1. ∫ x dx. n xn. = + C. + 1. ∫ dx = ln x + C. x. ∫. c dx = cx + C. x. 2.
4 ¤ CHAPTER 7 TECHNIQUES OF INTEGRATION 1 √ 1− 2 ,and arcsin √ 1− 2 sin .Toevaluatejustthelastintegral,nowlet = , =sin ⇒ = , = −cos .Thus,
Differentiation Formulas d dx k = 0 (1) d dx [f(x)±g(x)] = f0(x)±g0(x) (2) d dx [k ·f(x)] = k ·f0(x) (3) d dx [f(x)g(x)] = f(x)g0(x)+g(x)f0(x) (4) d dx f(x) ... −1 |x| √ x2 −1 (22) Integration Formulas Z dx = x+C (1) Z xn dx = xn+1 n+1 +C (2) Z dx x = ln|x|+C (3) Z ex dx = ex +C (4) Z ax dx = 1 lna ax +C (5) Z lnxdx = xlnx−x+C (6) Z
3 2 y = x − 6 x + 9 x − 4 , P ( 0, − 4 ) , R ( 4,0 ) ( x ) The figure above shows a curve with equation y = f ( x ) . The curve meets the x axis at the points P ( − 1,0 ) and Q , and its gradient function is given by. 8 x 3 − 1. ′ ( x ) = , x ≠ 0 . 2. Find an equation of the tangent to the curve at P .
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. 2. xndx xn+1. = + C. + 1. 3. dx = ln |x| + C.
Basic Integration Formulas. As with differentiation, there are two types of formulas, formulas for the integrals of specific functions and structural type formulas. Each formula for the derivative of a specific function corresponds to a formula for the derivative of an elementary function.
