Ihe atụ nke Ajụjụ Mkparịta ụka zuru oke
Integral integral bụ echiche dị mkpa na mgbakọ na mwepụ, nke e ji achọ ọrụ mbụ site na ọrụ e si n'aka onye mepụtara. A na-egosi integral integral site na akara ∫ nke na-esochi ya, ọrụ a ga-ejikọta na mgbanwe nke njikọta. N'isiokwu a, anyị ga-atụle ọtụtụ ihe atụ nke integral integral na ngwọta ha.
Ajụjụ Ihe atụ nke 1: Integral nke ọrụ Polynomial
Ajụjụ: Chọpụta ihe dị mkpa nke ọrụ ahụ \( f(x) = 3x^2 \).
Mkparịta ụka: Iji jikọta ọrụ polynomial, anyị na-eji iwu ndị bụ isi nke njikọta, ya bụ:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
Site n'iji iwu ndị a, ihe dị mkpa nke \(3x^2 \) bụ:
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \ekpe( \frac{1}{2+1} x^{2+1} \nri) + C = 3 \ekpe( \frac{1}{3} x^3 \nri) + C = x^3 + C \]
Ya mere, \( \int 3x^2 \, dx = x^3 + C \).
Ihe atụ Ajụjụ nke 2: Njikọta nke Ọrụ Exponential
Ajụjụ: Chọpụta ihe dị mkpa nke ọrụ ahụ \( f(x) = e^x \).
Mkparịta ụka: Njikọta nke ọrụ exponential \(e^x \) dị nnọọ mfe n'ihi na ọrụ \(e^x \) bụ ọrụ nke na-agbanwe agbanwe kpamkpam n'okpuru ọrụ dị iche na nke integral:
\[ \int e^x \, dx = e^x + C \]
Ya mere, \( \int e^x \, dx = e^x + C \).
Ihe atụ Ajụjụ nke 3: Integral nke ọrụ Trigonometric
Ajụjụ: Chọpụta ihe dị mkpa nke ọrụ ahụ \( f(x) = \sin(x) \).
Mkparịta ụka: Iji jikọta ọrụ trigonometric, anyị kwesịrị ịma ihe ndị bụ isi dị na ọrụ ndị ahụ. Otu n'ime mmekọrịta bụ isi bụ:
\[ \int \sin(x) \, dx = -\cos(x) + C \]
Ya mere, \( \int \sin(x) \, dx = -\cos(x) + C \).
Ihe atụ Ajụjụ nke 4: Njikọta nke Ọrụ Nkewa
Ajụjụ: Chọpụta ihe dị mkpa nke ọrụ ahụ \( f(x) = \frac{1}{x} \).
Mkparịta ụka: Ihe dị mkpa n'ọrụ ahụ bụ \( \frac{1}{x} \):
\[ \int \frac{1}{x} \, dx = \ln|x| +C\]
Ya mere, \( \int \frac{1}{x} \, dx = \ln|x| + C \).
Ihe atụ Ajụjụ nke 5: Njikọta nke Ọrụ Exponential Na-adịghị Mma
Ajụjụ: Chọpụta ihe dị mkpa nke ọrụ ahụ \( f(x) = x^{-2} \).
Mkparịta ụka: Maka \( n \neq -1 \), anyị na-eji iwu nke bụ isi:
\[ \int x^n \, dx = \frac{1}{n+1} x^{n+1} + C \]
N'okwu a, \(n = -2 \), yabụ:
\[ \int x^{-2} \, dx = \int x^{-2} \, dx = \frac{1}{-2+1} x^{-2+1} + C = \frac{1}{-1} x^{-1} + C = -x^{-1} + C = -\frac{1}{x} + C \]
Ya mere, \( \int x^{-2} \, dx = -\frac{1}{x} + C \).
Ihe atụ Ajụjụ nke 6: Njikọta nke Ọrụ Nchikota
Ajụjụ: Chọpụta ihe dị mkpa nke ọrụ ahụ \( f(x) = 4x^3 – 3x^2 + 2x – 5 \).
Mkparịta ụka: Anyị nwere ike ijikọta okwu ọ bụla iche iche site na iji iwu ndị bụ isi nke njikọta:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = \int 4x^3 \, dx – \int 3x^2 \, dx + \int 2x \, dx – \int 5 \, dx \]
Ugbu a, anyị na-ejikọta okwu ọ bụla n'otu n'otu:
\[ \int 4x^3 \, dx = 4 \int x^3 \, dx = 4 \ekpe( \frac{1}{3+1} x^{3+1} \nri) = 4 \ekpe( \frac{1}{4} x^4 \nri) = x^4 \]
\[ \int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \ekpe( \frac{1}{2+1} x^{2+1} \nri) = 3 \ekpe( \frac{1}{3} x^3 \nri) = x^3 \]
\[ \int 2x \, dx = 2 \int x \, dx = 2 \ekpe( \frac{1}{1+1} x^{1+1} \nri) = 2 \ekpe( \frac{1}{2} x^2 \nri) = x^2 \]
\[ \int 5 \, dx = 5x \]
Site na ijikọta nsonaazụ ndị a, anyị na-enweta:
\[ \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \]
Ya mere, \( \int (4x^3 – 3x^2 + 2x – 5) \, dx = x^4 – x^3 + x^2 – 5x + C \).
Mmechi
Indefinite integral bụ echiche dị oke mkpa na calculus ma nwee iwu dị iche iche nke na-eme ka ọ dịrị mfe ijikọ ụdị ọrụ dị iche iche. N'isiokwu a, anyị atụleela ọtụtụ ihe atụ nke indefinite integral, gụnyere polynomials, exponentials, trigonometric functions, fractions, functions with negative exponents, and combination of functions. Ịghọta na ịmụta iwu ndị a bụ isi nke integrals ga-enyere aka nke ukwuu n'idozi nsogbu calculus dị iche iche.
Njikọta ndị na-adịghị agwụ agwụ abụghị naanị ihe dị mkpa na tiori mgbakọ na mwepụ, kamakwa ha nwere ọtụtụ ojiji na fiziki, injinia, na ngalaba ndị ọzọ. Site na omume zuru oke, ijikọta ọrụ dị iche iche ga-adị mfe ma dịkwuo mfe nghọta.
