DẠNG 1. KHÁI NIỆM NGUYÊN HÀM, TÍNH CHẤT CỦA NGUYÊN HÀM

\({f^\prime }(x) = F(x)\forall x \in \mathbb{R}.\)

\({{\rm{f}}^\prime }({\rm{x}}) = {\rm{F}}({\rm{x}}) + {\rm{C}}({\rm{C}} \in \mathbb{R})\forall {\rm{x}} \in \mathbb{R}\)

\({F^\prime }(x) = f(x)\forall x \in \mathbb{R}.\)

\({F^\prime }(x) = f(x) + C(C \in \mathbb{R}\backslash \{ 0\} )\forall x \in \mathbb{R}.\)

\(\int f (x)dx = {f^\prime }(x) + C.\)

\(\int {{f^\prime }} (x)dx = f(x).\)

\(\int f (x)dx = {f^\prime }(x).\)

\(\int {{f^\prime }} (x)dx = f(x) + C.\)

\(\int {(f(} x) + g(x))dx = \int f (x)dx + \int g (x)dx\) với \(f(x),g(x)\) là hai hàm bất kì liên tục trên \(\mathbb{R}.\)

\(\int {({\rm{f}}(} {\rm{x}}) \cdot {\rm{g}}({\rm{x}})){\rm{dx}} = \int {\rm{f}} ({\rm{x}}){\rm{dx}} \cdot \int {\rm{g}} ({\rm{x}}){\rm{dx}}\) với \({\rm{f}}({\rm{x}}),{\rm{g}}({\rm{x}})\) là hai hàm bất kì liên tục trên \(\mathbb{R}.\)

\(\int {(f(} x) + g(x))dx = \int f (x)dx - \int g (x)dx\) với \(f(x),g(x)\) là hai hàm bất kì liên tục trên \(\mathbb{R}.\)

\(\int {\frac{{{\rm{f}}({\rm{x}})}}{{{\rm{g}}({\rm{x}})}}} {\rm{dx}} = \frac{{\int {\rm{f}} ({\rm{x}}){\rm{dx}}}}{{\int {\rm{g}} ({\rm{x}}){\rm{dx}}}}\) với \({\rm{f}}({\rm{x}}),{\rm{g}}({\rm{x}})\) là hai hàm bất kì liên tục trên \(\mathbb{R}.\)

\(\int {{\pi ^3}} {\rm{dx}} = \frac{{{\pi ^4}}}{4} + C.\)

\(\int {{\pi ^3}} {\rm{dx}} = 3{\pi ^2} + \) C.

\(\int {{\pi ^3}} {\rm{dx}} = 3{\pi ^2}.\)

\(\int {{\pi ^3}} dx = {\pi ^3}x + C.\)

\(\int 0 {\rm{dx}} = - {\rm{x}} + C.\)

\(\int 0 {\rm{dx}} = {\rm{x}} + C.\)

\(\int 0 {\rm{dx}} = {\rm{Cx}},{\rm{C}} \in \mathbb{R}\backslash \{ 0\} .\)

\(\int 0 {\rm{dx}} = {\rm{C}}.\)

\(\int k dx = x + C.\)

\(\int {{\rm{kdx}}} = {\rm{kx}} + {\rm{C}}.\)

\(\int k dx = C.\)

\(\int {{\rm{kdx}}} = {\rm{kx}}.\)C.

\({\rm{y}} = {{\rm{x}}^{\alpha + 1}}.\)

\(y = (\alpha + 1){x^{\alpha + 1}}.\)

\({\rm{y}} = \frac{{{{\rm{x}}^\alpha }}}{{\alpha + 1}}.\)

\(y = \frac{{{x^{\alpha + 1}}}}{{\alpha + 1}}.\)

\(\int {\sin } xdx = - \sin x + C.\)

\(\int {\sin } xdx = \sin x + C.\)

\(\int {\sin } xdx = - \cos x + C.\)

\(\int {\sin } xdx = \cos x + C.\)

\(\int {\cos } xdx = - \sin x + C.\)

\(\int {\cos } xdx = \sin x + C.\)

\(\int {\cos } xdx = - \cos x + C.\)

\(\int {\cos } xdx = \cos x + C.\)

\(\int {\frac{1}{{{{\sin }^2}x}}} dx = - \cot x + C.\)

\(\int {\frac{1}{{{{\sin }^2}x}}} dx = - \tan x + C.\)

\(\int {\frac{1}{{{{\sin }^2}x}}} dx = \cot x + C.\)

\(\int {\frac{1}{{{{\sin }^2}x}}} dx = \tan x + C.\)

\(\int {\frac{1}{{{{\cos }^2}x}}} dx = - \cot x + C.\)

\(\int {\frac{1}{{{{\cos }^2}x}}} dx = - \tan x + C.\)

\(\int {\frac{1}{{{{\cos }^2}x}}} dx = \cot x + C.\)

\(\int {\frac{1}{{{{\cos }^2}x}}} dx = \tan x + C.\)

\(\int {{{\rm{e}}^{\rm{x}}}} {\rm{dx}} = {{\rm{e}}^{ - {\rm{x}}}} + C.\)

\(\int {{{\rm{e}}^{\rm{x}}}} dx = {{\rm{e}}^{\rm{x}}} + C.\)

\(\int {{{\rm{e}}^{\rm{x}}}} {\rm{dx}} = - {{\rm{e}}^{\rm{x}}} + C.\)

\(\int {{{\rm{e}}^{\rm{x}}}} {\rm{dx}} = - {{\rm{e}}^{ - {\rm{x}}}} + C.\)

\(\int {{{\rm{a}}^{\rm{x}}}} {\rm{dx}} = \frac{{{{\rm{a}}^{\rm{x}}}}}{{\ln {\rm{a}}}} + C.\)

\(\int {{a^{\rm{x}}}} dx = {{\rm{a}}^{\rm{x}}}\ln {\rm{a}} + C.\)

\(\int {{{\rm{a}}^{\rm{x}}}} {\rm{dx}} = \frac{{{{\rm{e}}^{\rm{x}}}}}{{\ln {\rm{a}}}} + \) C.

\(\int {{{\rm{a}}^{\rm{x}}}} {\rm{dx}} = {{\rm{e}}^{\rm{x}}}\ln {\rm{a}} + C.\)

\(\int {\frac{1}{{\rm{x}}}} {\rm{dx}} = |{\rm{x}}| + {\rm{C}}.\)

\(\int {\frac{1}{{\rm{x}}}} {\rm{dx}} = \ln |{\rm{x}}| + {\rm{C}}.\)

\(\int {\ln } xdx = x + C.\)

\(\int {\ln } |x|dx = \ln x + C.\)

\({\rm{y}} = {{\rm{a}}^{\rm{x}}}.\)

\({\rm{y}} = {{\rm{a}}^{{\rm{x}} + 1}}.\)

\(y = \frac{{{a^x}}}{{\ln a}}.\)

\({\rm{y}} = {{\rm{a}}^{\rm{x}}}\ln {\rm{a}}.\)

\({\rm{y}} = \frac{1}{{{\rm{x}}\ln {\rm{e}}}}.\)

\(y = \frac{1}{{x\ln a}}.\)

\(y = \frac{1}{x}.\)

\(y = \frac{{\ln {\rm{a}}}}{{\rm{x}}}.\)

\({\rm{y}} = {\rm{a}}\cos ({\rm{ax}} + {\rm{b}}).\)

\({\rm{y}} = \frac{{ - \cos ({\rm{ax}} + {\rm{b}})}}{{\rm{a}}}\)

\({\rm{y}} = - {\rm{a}}\cos ({\rm{ax}} + {\rm{b}}).\)

\({\rm{y}} = \frac{{\cos ({\rm{ax}} + {\rm{b}})}}{{\rm{a}}}.\)

\(y = - a\sin (ax + b).\)

\({\rm{y}} = \frac{{ - \sin ({\rm{ax}} + {\rm{b}})}}{{\rm{a}}}.\)

\({\rm{y}} = {\rm{a}}\sin ({\rm{ax}} + {\rm{b}}).\)

\({\rm{y}} = \frac{{\sin ({\rm{ax}} + {\rm{b}})}}{{\rm{a}}}.\)

\({\rm{y}} = \frac{1}{{{\rm{ax}} + {\rm{b}}}}.\)

\(y = \frac{{ - 1}}{{{\rm{ax}} + {\rm{b}}}}.\)

\(y = \frac{{ - {\rm{a}}}}{{{\rm{ax}} + {\rm{b}}}}.\)

\({\rm{y}} = \frac{{\rm{a}}}{{{\rm{ax}} + {\rm{b}}}}.\)

\({\rm{y}} = - {{\rm{e}}^{{\rm{ax}} + {\rm{b}}}}.\)

\({\rm{y}} = {{\rm{e}}^{{\rm{ax}} + {\rm{b}}}}.\)

\(y = a{e^{ax + b}}.\)

\(y = \frac{{{e^{ax + b}}}}{a}\)

25.

625.

5.

125.

\(f(x) = \sin x.\)

\(f(x) = - \cos x.\)

\(f(x) = - \sin x.\)

\(f(x) = \cos x.\)

\(y = \cos 2x.\)

\({\rm{y}} = \frac{{\cos 2{\rm{x}}}}{2}.\)

\(y = 2\cos 2x.\)

\(y = \frac{{ - \cos 2x}}{2}.\)

\(y = \cos 2x.\)

\({\rm{y}} = \frac{{\cos 2{\rm{x}}}}{2}.\)

\(y = 2\cos 2x.\)

\(y = \frac{{ - \cos 2x}}{2}.\)

\({\rm{y}} = {{\rm{x}}^6}.\)

\({\rm{y}} = 5{{\rm{x}}^4}.\)

\({\rm{y}} = \frac{{{{\rm{x}}^6}}}{6}.\)

\({\rm{y}} = 6{{\rm{x}}^5}.\)

\({\rm{y}} = \frac{{{{\rm{x}}^4}}}{4} + 1\)

\(y = \frac{{{x^4}}}{4} + 2\)

\({\rm{y}} = \frac{{{{\rm{x}}^4}}}{4} + 3.\)

\({\rm{y}} = 3{{\rm{x}}^2}.\)

\(\frac{{{x^2}}}{2} - x - \ln |x| + C\)

\(\frac{{{x^2}}}{2} - x + \ln |x| + C.\)

\(\frac{{{x^2}}}{2} + x - \ln |x| + C.\)

\({x^2} - x - \ln |x| + C.\)

\(\int {\frac{{{\rm{dx}}}}{{{{\rm{x}}^3}}}} = \frac{1}{{{{\rm{x}}^2}}} + C.\)

\(\int {\frac{{dx}}{{{x^3}}}} = \frac{2}{{{x^2}}} + C.\)

\(\int {\frac{{dx}}{{{x^3}}}} = \frac{{ - 1}}{{2{x^2}}} + C.\)

\(\int {\frac{{dx}}{{{x^3}}}} = \frac{{ - 2}}{{{x^2}}} + C.\)

\(\int {\frac{{{\rm{dx}}}}{{\sqrt {{{\rm{x}}^3}} }}} = \frac{1}{{\sqrt {\rm{x}} }} + C.\)

\(\int {\frac{{{\rm{dx}}}}{{\sqrt {{{\rm{x}}^3}} }}} = \frac{2}{{3\sqrt {\rm{x}} }} + \) C.

\(\int {\frac{{dx}}{{\sqrt {{{\rm{x}}^3}} }}} = \frac{{ - 2}}{{\sqrt {\rm{x}} }} + C.\)

\(\int {\frac{{{\rm{dx}}}}{{\sqrt {{{\rm{x}}^3}} }}} = \frac{2}{{\sqrt {\rm{x}} }} + C.\)

\( - \cos x + x + C.\)

\( - \cos x - x + C.\)

\(\cos x + x + C.\)

\(\cos x - x + C.\)

\(\frac{{x + \sin x}}{2} + C.\)

\(\frac{{x - \sin x}}{2} + C.\)

\(\frac{{x - \cos x}}{2} + C.\)

\(\frac{{x + \cos x}}{2} + C.\)

\(\frac{{x + \sin x}}{2} + C.\)

\(\frac{{x - \sin x}}{2} + C.\)

\(\frac{{x - \cos x}}{2} + C.\)

\(\frac{{x + \cos x}}{2} + C.\)

\({\rm{y}} = \frac{{{{\tan }^3}{\rm{x}}}}{3}.\)

\(y = \tan x - x.\)

\(y = - \tan x + x.\)

\(y = \tan x.\)

\({\rm{y}} = \frac{{{{\cot }^3}{\rm{x}}}}{3}.\)

\(y = \cot x - x.\)

\(y = \cot x.\)

\(y = - \cot x - x.\)

\( - \frac{1}{{25}} \cdot \frac{{{{(0,4)}^{\rm{x}}}}}{{\ln 0,4}} + \frac{3}{{25}} \cdot \frac{{{{(0,6)}^{\rm{x}}}}}{{\ln 0,6}} + C.\)

\(\frac{1}{{25}} \cdot \frac{{{{(0,4)}^{\rm{x}}}}}{{\ln 0,4}} + \frac{3}{{25}} \cdot \frac{{{{(0,6)}^{\rm{x}}}}}{{\ln 0,6}} + C.\)

\( - \frac{1}{{25}} \cdot \frac{{{{(0,4)}^{\rm{x}}}}}{{\ln 0,4}} - \frac{3}{{25}} \cdot \frac{{{{(0,6)}^{\rm{x}}}}}{{\ln 0,6}} + C.\)

\(\frac{1}{{25}} \cdot \frac{{{{(0,4)}^{\rm{x}}}}}{{\ln 0,4}} - \frac{3}{{25}} \cdot \frac{{{{(0,6)}^{\rm{x}}}}}{{\ln 0,6}} + C.\)

\(\int {{e^{ax + b}}} dx = {e^b}\int {{{\left( {{e^a}} \right)}^x}} dx = {e^b} \cdot \frac{{{{\left( {{e^a}} \right)}^x}}}{{\ln {e^a}}} + C = \frac{{{e^{ax + b}}}}{a} + C.\)

\(\int {{e^{ax + b}}} dx = {e^b}\int {{{\left( {{e^a}} \right)}^x}} dx = {e^b} \cdot {\left( {{e^a}} \right)^x} + C = {e^{ax + b}} + C.\)

\(\int {{e^{ax + b}}} dx = {e^b}\int {{{\left( {{e^a}} \right)}^x}} dx = {e^b} \cdot \frac{{{{\left( {{e^a}} \right)}^x}}}{{\ln {e^a}}} + C = \frac{{{e^{bx + a}}}}{a} + C.\)

\(\int {{e^{ax + b}}} dx = {e^b}\int {{{\left( {{e^a}} \right)}^x}} dx = {e^b} \cdot \frac{{{{\left( {{e^a}} \right)}^x}}}{{\ln {{\rm{e}}^a}}} + C = \frac{{{e^{ax + b}}}}{b} + C.\)