Rules of Integrals with Examples
A tutorial, with examples and detailed solutions, in using the rules of indefinite integrals in calculus is presented. A set of questions with solutions is also included.
In what follows, C is a constant of integration and can take any value.
1 - Integral of a power function: f(x) = x ^{n}Example: Evaluate the integral Solution:
2 - Integral of a function f multiplied by a constant k: k f(x)Example: Evaluate the integral Solution: According to the above rule ∫5 sin (x) dx = 5∫sin(x) dx ∫sin(x) dx is given by 2.1 in table of integral formulas, hence ∫5 sin(x) dx = - 5 cos x + C
3 - Integral of Sum of Functions.Example: Evaluate the integral Solution: According to the above property ∫[x + e^{ x}] dx = ∫x dx + ∫e^{ x} dx ò x dx is given by 1.3 and ∫e^{ x} dx by 4.1 in table of integral formulas, hence ∫[x + e^{ x}] dx = x^{ 2} / 2 + e^{ x} + c
4 - Integral of Difference of Functions.Example: Evaluate the integral Solution: According to the above property ∫[2 - 1/x] dx = ∫2 dx - ∫(1/x) dx ∫2 dx is given by 1.2 and ∫(1/x) dx by 1.4 in table of integral formulas, hence ∫[2 - 1/x] dx = 2x - ln |x| + c
5 - Integration by Substitution.Example: Evaluate the integral Solution: Let u = x^{ 2} - 1, du/dx = 2x and the given integral can be written as ∫(x^{ 2} - 1)^{ 20} 2x dx = ∫u^{ 20} (du/dx) dx = ∫u^{ 20} du according to above property = u^{ 21} / 21 + c = (x^{ 2} - 1)^{ 21} / 21 + c
6 - Integration by Parts.Example: Evaluate the integral Solution: Let f(x) = x and g ' (x) = cos x which gives f ' (x) = 1 and g(x) = sin x From integration by parts formula above, ∫x cos x dx = x sin x - ∫1 sin x dx = x sin x + cos x + c
More Questions with SolutionsUse the table of integral formulas and the rules above to evaluate the following integrals. [Note that you may need to use more than one of the above rules for one integral].1. ∫(1 / 2) ln (x) dx 2. ∫[sin (x) + x^{ 5}] dx 3. ∫[sinh (x) - 3] dx 4. ∫ - x sin (x) dx 5. ∫ sin ^{10}(x) cos(x) dx Solutions to the Above Questions1. This is the integral of ln (x) multiplied by 1 / 2 and we therefore use rule 2 above to obtain: ∫(1 / 2) ln (x) dx = (1 / 2) ∫ln (x) dx We now use formula 4.3 in the table of integral formulas to evaluate ∫ln (x) dx. Hence ∫(1 / 2) ln (x) dx = (1 / 2) ( (x ln (x)) - x ) + c 2. Use rule 3 ( integral of a sum ) to obtain ∫[sin (x) + x^{ 5}] dx = ∫ sin (x) dx + ∫x^{ 5} dx We use formula 2.1 in the table of integral formulas to evaluate ∫ sin (x) dx and rule 1 above to evaluate ∫x^{ 5} dx. Hence ∫[sin (x) + x^{ 5}] dx = - cos (x) + x ^{ 6} / 6 3. Use rule 4 (integral of a difference) to obtain ∫(sinh (x) - 3) dx = ∫ sinh (x) dx - ∫3 dx We use formula 7.1 in the table of integral formulas to evaluate ∫ sinh (x) dx and integral of the constant 3 to obtain ∫(sinh (x) - 3) dx = cosh (x) - 3 x + c 4. The integrand is the product of two function x and sin (x) and we try to use integration by parts in rule 6 as follows: Let f(x) = x , g'(x) = sin(x) and therefore g(x) = - cos(x) Hence ∫ - x sin (x) dx = - ∫ f(x) g'(x) dx = - ( f(x) g(x) - ∫ f'(x) g(x) dx) Substitute f(x), f'(x), g(x) and g'(x) by x , 1, sin(x) and - cos(x) respectively to write the integral as = - x (- cos(x)) + ∫ 1 (- cos(x)) dx Use formula 2.2 in in the table of integral formulas to evaluate ∫ cos(x) dx and simplify to obtain = x cos (x) - sin(x) + c 5. Let u = sin(x) and therefore du/dx = cos(x). Hence the given integral can be written as ∫ sin^{10}(x) cos dx = ∫( u^{10} du/dx ) dx Use rule 5 to write = ∫ u^{10} du which gives = u ^{11} / 11 + c Substitute u by sin(x) to obtain = (1 / 11) (sin ^{11}(x) ) + c More References and LinksTable of Integral Formulasintegrals and their applications in calculus. evaluate integrals. Integration by Substitution. |