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Antiderivatives and the Indefinite Integral. The most antiderivatives we know is derived from the table of derivatives, which we read in the opposite direction. 8v9=4 9 + 24v5=4 5 v 3 + C 10. v6 2 3v8=3 8 +C 11. The Teaching & Learning Plans . ANSWERS Inde nite integrals: 1. 4z 6 6 + 7z 3 3 + z2 2 +C 7. Example 2: Compute . are structured as follows: Aims. SECTIONS 5.1 & 5.2: ANTIDERIVATIVES AND INDEFINITE INTEGRALS 5 EXERCISES Find the following integrals. • Use anti-differentiation to solve real world problems in which . ©9 x280 z1537 TK su HtQaY tS 2o XfxtRw ka 1rRe v eLXLBCl. 2u3=2 +2u1=2 +C 8. The Indefinite Integral and Basic Rules of Integration. 3x3 3x2 +x+C 12. x3 3 2x x 41. cot1 +C 13. Find Z 9x3 + 8x2 + 3x 4 3x3 dx. Calculation of integrals using the linear properties of indefinite integrals and the table of basic integrals is called direct integration… See Figure 8.1. Notation: Integration and Indefinite Integral The fact that the set of functions F(x) + C represents all antiderivatives of f (x) is denoted by: ∫f(x)dx=F(x)+C where the symbol ∫ is called the integral sign, f (x) is the integrand, C is the constant of integration, and dx denotes the independent variable we are integrating with respect to. 5.5: Indefinite Integrals and the Substitution Rule Last updated; Save as PDF ... (when one or both of the limits of integration are variables). 3t3 2t2 +3t+C 4. t4 2 t3 3 + 3t2 2 7t+C 5. z 2 2 +3z 21 +C 6. 2u5=2 5 + u 1 2 +5u+C 9. INDEFINITE INTEGRALS Example 6. ... • Find the indefinite form of the anti-derivative of a function. M f 1M Fa5d oep 2w Ti 8t ahf 9I in7f vignQift BeD VCfa il ec uyl 7u jsP.W Worksheet by Kuta Software LLC Given these rules together with Theorem 4.1, we will be able to solve a great variety of definite integrals. Table of basic integrals $$\int dx = x + C$$ $$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n eq 1$$ $$\int \frac{1}{x} dx = \ln |x| + C$$ Compute the following indefinite integral. Integrals with Trigonometric Functions Z sinaxdx= 1 a cosax (63) Z sin2 axdx= x 2 sin2ax 4a (64) Z sinn axdx= 1 a cosax 2F 1 1 2; 1 n 2; 3 2;cos2 ax (65) Z sin3 axdx= 3cosax 4a + cos3ax 12a (66) Z cosaxdx= 2x3 3 1. Integral Calculus. But these integrals are very similar geometrically . Solution: Using our rules we have Sometimes our rules need to be modified slightly due to operations with constants as is the case in the following example. Thus, y = x2 + C, where C is arbitrary constant, represents a family of integrals. Example 7. Leaving Certificate Syllabus. 4x3 3 4x2 +x+C 3. Solution: Lesson Summary Integration by Parts Recall the Product Rule: d dx [u(x)v(x)] = v(x) du dx + u(x) dv dx 2. Then apply the Power Rule and the Arcsine Rule as follows. By assigning dif ferent values to C, we get dif ferent members of the family . • Find a distinct anti-derivative of a function. These together constitute the indefinite integral. Find Z 3 x + e2x + 5e 4x 7e3x dx. An indefinite integral represents a family of functions, all of which differ by a constant. O 4 KAnl UlI RrPi rg ChAtNs8 trFe KseUrNvOeOd1. Find Z x2 5x+ 2 x dx. Z integrals. 2x2 +3x+C 2. Solution: Example 3: Compute . Check your answer by di erentiating. We conclude the lesson by stating the rules for definite integrals, most of which parallel the rules we stated for the general indefinite integrals. SECTION 8.1 Basic Integration Rules 519 EXAMPLE 2 Using Two Basic Rules to Solve a Single Integral Evaluate Solution Begin by writing the integral as the sum of two integrals. Integrating both sides and solving for one of the integrals leads to our Integration by Parts formula: Z udv= uv Z vdu Integration by Parts (which I may abbreviate as IbP or IBP) \undoes" the Product Rule. 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