All integration formulas

Common Functions Function Integral Constant ∫a dx ax + C Variable ∫x dx x 2/2 + C Square ∫x 2 dx x 3/3 + C Reciprocal ∫(1/x) dx ln|x| + C Exponential ∫e x dx e x + C ∫a x dx a x/ln(a) + C ∫ln(x) dx x ln(x) − x + C Trigonometry (x in ∫cos(x) dx sin(x) + C ∫sin(x) dx -cos(x) + C ∫sec 2(x) dx tan(x) + C Rules Function Integral Multiplication by constant ∫cf(x) dx c ∫f(x) dx Power Rule (n≠−1) ∫x n dx x n+1 n+1 + C Sum Rule ∫(f + g) dx ∫f dx + ∫g dx Difference Rule ∫(f - g) dx ∫f dx - ∫g dx Integration by Parts See Substitution Rule See Example: What is ∫(8z + 4z 3 − 6z 2) dz ? Use the Sum and Difference Rule: ∫(8z + 4z 3 − 6z 2) dz = ∫8z dz + ∫4z 3 dz − ∫6z 2 dz Constant Multiplication: = 8 ∫z dz + 4 ∫z 3 dz − 6 ∫z 2 dz Power Rule: = 8z 2/2 + 4z 4/4 − 6z 3/3 + C Simplify: = 4z 2 + z 4 − 2z 3 + C Integration by Parts See Substitution Rule See Final Advice • Get plenty of practice • Don't forget the dx (or dz, etc) • Don't forget the + C

Basic Integrals 1. ∫ u n d u = u n + 1 n + 1 + C , n ≠ − 1 ∫ u n d u = u n + 1 n + 1 + C , n ≠ − 1 2. ∫ d u u = ln | u | + C ∫ d u u = ln | u | + C 3. ∫ e u d u = e u + C ∫ e u d u = e u + C 4. ∫ a u d u = a u ln a + C ∫ a u d u = a u ln a + C 5. ∫ sin u d u = −cos u + C ∫ sin u d u = −cos u + C 6. ∫ cos u d u = sin u + C ∫ cos u d u = sin u + C 7. ∫ sec 2 u d u = tan u + C ∫ sec 2 u d u = tan u + C 8. ∫ csc 2 u d u = −cot u + C ∫ csc 2 u d u = −cot u + C 9. ∫ sec u tan u d u = sec u + C ∫ sec u tan u d u = sec u + C 10. ∫ csc u cot u d u = −csc u + C ∫ csc u cot u d u = −csc u + C 11. ∫ tan u d u = ln | sec u | + C ∫ tan u d u = ln | sec u | + C 12. ∫ cot u d u = ln | sin u | + C ∫ cot u d u = ln | sin u | + C 13. ∫ sec u d u = ln | sec u + tan u | + C ∫ sec u d u = ln | sec u + tan u | + C 14. ∫ csc u d u = ln | csc u − cot u | + C ∫ csc u d u = ln | csc u − cot u | + C 15. ∫ d u a 2 − u 2 = sin −1 u a + C ∫ d u a 2 − u 2 = sin −1 u a + C 16. ∫ d u a 2 + u 2 = 1 a tan −1 u a + C ∫ d u a 2 + u 2 = 1 a tan −1 u a + C 17. ∫ d u u u 2 − a 2 = 1 a sec −1 u a + C ∫ d u u u 2 − a 2 = 1 a sec −1 u a + C Trigonometric Integrals 18. ∫ sin 2 u d u = 1 2 u − 1 4 sin 2 u + C ∫ sin 2 u d u = 1 2 u − 1 4 sin 2 u + C 19. ∫ cos 2 u d u = 1 2 u + 1 4 sin 2 u + C ∫ cos 2 u d u = 1 2 u + 1 4 sin 2 u + C 20. ∫ tan 2 u d u = tan u − u + C ∫ tan 2 u d u = tan u − u + C 21. ∫ cot 2 u d u = − cot u − u + C ∫ cot 2 u d u = − cot u − u + C 22. ∫ sin 3 u d u = − 1 3 ( 2 + sin 2 u ) cos u + C ∫ sin ...

Integration Techniques Many integration formulas can be derived directly from their corresponding derivative formulas, while other integration problems require more work. Some that require more work are substitution and change of variables, integration by parts, trigonometric integrals, and trigonometric substitutions. Basic formulas Most of the following basic formulas directly follow the differentiation rules. • • • • • • • • • • • • • • • • • • • • Example 1: Evaluate Using formula (4) from the preceding list, you find that . Example 2: Evaluate . Because using formula (4) from the preceding list yields Example 3: Evaluate Applying formulas (1), (2), (3), and (4), you find that Example 4: Evaluate Using formula (13), you find that Example 5: Evaluate Using formula (19) with a = 5, you find that Substitution and change of variables One of the integration techniques that is useful in evaluating indefinite integrals that do not seem to fit the basic formulas is substitution and change of variables. This technique is often compared to the chain rule for differentiation because they both apply to composite functions. In this method, the inside function of the composition is usually replaced by a single variable (often u). Note that the derivative or a constant multiple of the derivative of the inside function must be a factor of the integrand. The purpose in using the substitution technique is to rewrite the integration problem in terms of the new variable so that one or mor...

Integration Rules Integration rules: Integration is used to find many useful parameters or quantities like area, volumes, central points, etc., on a large scale. The most common application of integration is to find the area under the curve on a graph of a function. To work out the integral of more complicated functions than just the known ones, we have some integration rules. These rules can be studied below. Apart from these rules, there are many Integration Rules of Basic Functions The integration rules are defined for different types of functions. Let us learn here the basic rules for integration of the some common functions, such as: • Constant • Variable • Square • Reciprocal • Exponential • Trigonometry Integration of Constant Integration of constant function say ‘a’ will result in: ∫a dx = ax + C Example: ∫4 dx = 4x + C Integration of Variable If x is any variable then; ∫x dx = x 2 /2 + C Integration of Square If the given function is a square term, then; ∫x 2 dx = x 3 /3 Integration of Reciprocal If 1/x is a reciprocal function of x, then the integration of this function is: ∫(1/x) dx = ln|x| + C        (Natural log of x) Integration of Exponential Function The different rules for integration of exponential functions are: • ∫e x dx = e x + C • ∫a x dx = a x /ln(a) + C • ∫ln(x) dx = x ln(x) − x + C Integration of Trigonometric Function • ∫cos(x) dx = sin(x) + C • ∫sin(x) dx = -cos(x) + C • ∫sec 2 (x) dx = tan(x) + C Important In...

Integral Calculus Integral calculus helps in findingthe anti-derivatives of a function. These anti-derivatives are also called the integralsof the function.The process of finding the anti-derivative of a function is called integration. The inverse process of finding derivatives is finding the integrals. The integral of a function represents a family of curves. Finding both derivatives and integrals form the fundamental calculus. In this topic, we will cover the basics of integrals and evaluating integrals. 1. 2. 3. 4. 5. 6. 7. 8. What isIntegral Calculus? Integrals are the values of the function found by the process of Example: Given: f(x) = x 2 . Derivative of f(x) = f'(x) = 2x = g(x) if g(x) = 2x, thenanti-derivative of g(x) = ∫ g(x) = x 2 Definition of Integral F(x) is called an antiderivative or Newton-Leibnitz integral or primitive of a function f(x) on an interval I. F'(x) = f(x), for every value of x in I. Integral is the representation of the area of a region under a curve. We approximate the actual value of an integral by drawing rectangles. A Fundamental Theorems of Integral Calculus We define integrals as the function of the area bounded by the curve y = f(x), a ≤ x ≤ b, the x-axis, and the ordinates x = a and x =b, where b>a. Let x be a given point in [a,b]. Then\(\int\limits_a^b f(x) dx\) represents the area function. This concept of area function leads to the fundamental theorems of integral calculus. • First Fundamental Theorem of Integral Calculus • Second ...

Integration Rules Integration rules: Integration is used to find many useful parameters or quantities like area, volumes, central points, etc., on a large scale. The most common application of integration is to find the area under the curve on a graph of a function. To work out the integral of more complicated functions than just the known ones, we have some integration rules. These rules can be studied below. Apart from these rules, there are many Integration Rules of Basic Functions The integration rules are defined for different types of functions. Let us learn here the basic rules for integration of the some common functions, such as: • Constant • Variable • Square • Reciprocal • Exponential • Trigonometry Integration of Constant Integration of constant function say ‘a’ will result in: ∫a dx = ax + C Example: ∫4 dx = 4x + C Integration of Variable If x is any variable then; ∫x dx = x 2 /2 + C Integration of Square If the given function is a square term, then; ∫x 2 dx = x 3 /3 Integration of Reciprocal If 1/x is a reciprocal function of x, then the integration of this function is: ∫(1/x) dx = ln|x| + C        (Natural log of x) Integration of Exponential Function The different rules for integration of exponential functions are: • ∫e x dx = e x + C • ∫a x dx = a x /ln(a) + C • ∫ln(x) dx = x ln(x) − x + C Integration of Trigonometric Function • ∫cos(x) dx = sin(x) + C • ∫sin(x) dx = -cos(x) + C • ∫sec 2 (x) dx = tan(x) + C Important In...

Integration Techniques Many integration formulas can be derived directly from their corresponding derivative formulas, while other integration problems require more work. Some that require more work are substitution and change of variables, integration by parts, trigonometric integrals, and trigonometric substitutions. Basic formulas Most of the following basic formulas directly follow the differentiation rules. • • • • • • • • • • • • • • • • • • • • Example 1: Evaluate Using formula (4) from the preceding list, you find that . Example 2: Evaluate . Because using formula (4) from the preceding list yields Example 3: Evaluate Applying formulas (1), (2), (3), and (4), you find that Example 4: Evaluate Using formula (13), you find that Example 5: Evaluate Using formula (19) with a = 5, you find that Substitution and change of variables One of the integration techniques that is useful in evaluating indefinite integrals that do not seem to fit the basic formulas is substitution and change of variables. This technique is often compared to the chain rule for differentiation because they both apply to composite functions. In this method, the inside function of the composition is usually replaced by a single variable (often u). Note that the derivative or a constant multiple of the derivative of the inside function must be a factor of the integrand. The purpose in using the substitution technique is to rewrite the integration problem in terms of the new variable so that one or mor...

Integral Calculus Integral calculus helps in findingthe anti-derivatives of a function. These anti-derivatives are also called the integralsof the function.The process of finding the anti-derivative of a function is called integration. The inverse process of finding derivatives is finding the integrals. The integral of a function represents a family of curves. Finding both derivatives and integrals form the fundamental calculus. In this topic, we will cover the basics of integrals and evaluating integrals. 1. 2. 3. 4. 5. 6. 7. 8. What isIntegral Calculus? Integrals are the values of the function found by the process of Example: Given: f(x) = x 2 . Derivative of f(x) = f'(x) = 2x = g(x) if g(x) = 2x, thenanti-derivative of g(x) = ∫ g(x) = x 2 Definition of Integral F(x) is called an antiderivative or Newton-Leibnitz integral or primitive of a function f(x) on an interval I. F'(x) = f(x), for every value of x in I. Integral is the representation of the area of a region under a curve. We approximate the actual value of an integral by drawing rectangles. A Fundamental Theorems of Integral Calculus We define integrals as the function of the area bounded by the curve y = f(x), a ≤ x ≤ b, the x-axis, and the ordinates x = a and x =b, where b>a. Let x be a given point in [a,b]. Then\(\int\limits_a^b f(x) dx\) represents the area function. This concept of area function leads to the fundamental theorems of integral calculus. • First Fundamental Theorem of Integral Calculus • Second ...

Basic Integrals 1. ∫ u n d u = u n + 1 n + 1 + C , n ≠ − 1 ∫ u n d u = u n + 1 n + 1 + C , n ≠ − 1 2. ∫ d u u = ln | u | + C ∫ d u u = ln | u | + C 3. ∫ e u d u = e u + C ∫ e u d u = e u + C 4. ∫ a u d u = a u ln a + C ∫ a u d u = a u ln a + C 5. ∫ sin u d u = −cos u + C ∫ sin u d u = −cos u + C 6. ∫ cos u d u = sin u + C ∫ cos u d u = sin u + C 7. ∫ sec 2 u d u = tan u + C ∫ sec 2 u d u = tan u + C 8. ∫ csc 2 u d u = −cot u + C ∫ csc 2 u d u = −cot u + C 9. ∫ sec u tan u d u = sec u + C ∫ sec u tan u d u = sec u + C 10. ∫ csc u cot u d u = −csc u + C ∫ csc u cot u d u = −csc u + C 11. ∫ tan u d u = ln | sec u | + C ∫ tan u d u = ln | sec u | + C 12. ∫ cot u d u = ln | sin u | + C ∫ cot u d u = ln | sin u | + C 13. ∫ sec u d u = ln | sec u + tan u | + C ∫ sec u d u = ln | sec u + tan u | + C 14. ∫ csc u d u = ln | csc u − cot u | + C ∫ csc u d u = ln | csc u − cot u | + C 15. ∫ d u a 2 − u 2 = sin −1 u a + C ∫ d u a 2 − u 2 = sin −1 u a + C 16. ∫ d u a 2 + u 2 = 1 a tan −1 u a + C ∫ d u a 2 + u 2 = 1 a tan −1 u a + C 17. ∫ d u u u 2 − a 2 = 1 a sec −1 u a + C ∫ d u u u 2 − a 2 = 1 a sec −1 u a + C Trigonometric Integrals 18. ∫ sin 2 u d u = 1 2 u − 1 4 sin 2 u + C ∫ sin 2 u d u = 1 2 u − 1 4 sin 2 u + C 19. ∫ cos 2 u d u = 1 2 u + 1 4 sin 2 u + C ∫ cos 2 u d u = 1 2 u + 1 4 sin 2 u + C 20. ∫ tan 2 u d u = tan u − u + C ∫ tan 2 u d u = tan u − u + C 21. ∫ cot 2 u d u = − cot u − u + C ∫ cot 2 u d u = − cot u − u + C 22. ∫ sin 3 u d u = − 1 3 ( 2 + sin 2 u ) cos u + C ∫ sin ...

Common Functions Function Integral Constant ∫a dx ax + C Variable ∫x dx x 2/2 + C Square ∫x 2 dx x 3/3 + C Reciprocal ∫(1/x) dx ln|x| + C Exponential ∫e x dx e x + C ∫a x dx a x/ln(a) + C ∫ln(x) dx x ln(x) − x + C Trigonometry (x in ∫cos(x) dx sin(x) + C ∫sin(x) dx -cos(x) + C ∫sec 2(x) dx tan(x) + C Rules Function Integral Multiplication by constant ∫cf(x) dx c ∫f(x) dx Power Rule (n≠−1) ∫x n dx x n+1 n+1 + C Sum Rule ∫(f + g) dx ∫f dx + ∫g dx Difference Rule ∫(f - g) dx ∫f dx - ∫g dx Integration by Parts See Substitution Rule See Example: What is ∫(8z + 4z 3 − 6z 2) dz ? Use the Sum and Difference Rule: ∫(8z + 4z 3 − 6z 2) dz = ∫8z dz + ∫4z 3 dz − ∫6z 2 dz Constant Multiplication: = 8 ∫z dz + 4 ∫z 3 dz − 6 ∫z 2 dz Power Rule: = 8z 2/2 + 4z 4/4 − 6z 3/3 + C Simplify: = 4z 2 + z 4 − 2z 3 + C Integration by Parts See Substitution Rule See Final Advice • Get plenty of practice • Don't forget the dx (or dz, etc) • Don't forget the + C