Bảng công thức Nguyên hàm từ cơ bản đến nâng cao

(i) \(\int \frac{1}{x^{2}+a^{2}} d x=\frac{1}{a} \tan ^{-1} \frac{x}{a}+C\)

(ii) \(\int \frac{1}{a^{2}-x^{2}} d x=\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+C=\frac{1}{a} \tanh ^{-1}\left(\frac{x}{a}\right)+C\)

(iii) \(\int \frac{d x}{x^{2}-a^{2}} d x=\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+C=-\frac{1}{a} \operatorname{coth}^{-1}\left(\frac{x}{a}\right)+C\)

(iv) \(\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\sin ^{-1} \frac{x}{a}+C\)

(v) \(\int \frac{d x}{\sqrt{x^{2}-a^{2}}}=\log \left|x+\sqrt{x^{2}-a^{2}}\right|+C=\cosh ^{-1}\left(\frac{x}{a}\right)+C\)

(vi) \(\left.\int \frac{d x}{\sqrt{x^{2}+a^{2}}}=\log 1 x+\sqrt{x^{2}+a^{2}}\right)+C=\sinh ^{-1}\left(\frac{x}{a}\right)+C\)

(vii) \(\int \sqrt{x^{2}+a^{2}} d x=\frac{1}{2}\left[x \sqrt{x^{2}+a^{2}}+a^{2} \log \left(x+\sqrt{x^{2}+a^{2} \mid}\right]+C\right.\)(viii) \(\left[\sqrt{a^{2}-x^{2}} d x=\frac{1}{2}\left[x \sqrt{a^{2}-x^{2}}+a^{2} \sin ^{-1}\left(\frac{x}{a}\right)\right]+C\right.\)

(ix) \(\int \sqrt{x^{2}-a^{2}} d x=\frac{1}{2}\left[x \sqrt{x^{2}-a^{2}}-a^{2} \log \left|x+\sqrt{x^{2}-a^{2}}\right|\right]+C\)

(x) \(\int(p x+q) \sqrt{a x^{2}+b x+c} d x=\frac{p}{2 a} \int(2 a x+b) \sqrt{a x^{2}+b x+c} d x\)

(1) \(\int(a x+b)^{\alpha} \mathrm{d} x=\frac{1}{a} \frac{(a x+b)^{\alpha+1}}{\alpha+1}+C(\alpha \neq-1)\)

(2) \(\int \mathrm{e}^{a x+b} \mathrm{~d} x=\frac{1}{a} \mathrm{e}^{a x+b}+C\)

(3) \(\int \sin (a x+b) \mathrm{d} x=-\frac{1}{a} \cos (a x+b)+C\)

(4) \(\int \frac{1}{\sin ^{2}(a x+b)} \mathrm{d} x=-\frac{1}{a} \cot (a x+b)+C\)

(5) \(\int \cot (a x+b) \mathrm{d} x=\frac{1}{a} \ln |\sin (a x+b)|+C\)

(6) \(\int \frac{\mathrm{d} x}{a^{2}-x^{2}}=\frac{1}{2 a} \ln \left|\frac{a+x}{a-x}\right|+C\)

(7) \(\int \frac{\mathrm{d} x}{\sqrt{a^{2}-x^{2}}}=\arcsin \frac{x}{|a|}=C\)

(8) \(\int \ln (a x+b) \mathrm{d} x=\left(x+\frac{b}{a}\right) \ln (a x+b)-x+C\)

(9) \(\int \mathrm{e}^{a x} \cos b x \mathrm{~d} x=\frac{\mathrm{e}^{a x}(a \cos b x)+b \sin b x}{a^{2}+b^{2}}+C\)

(10) \(\int \frac{1}{a x+b} \mathrm{~d} x=\frac{1}{a} \ln |a x+b|+C\)

(11) \(\int \cos (a x+b) \mathrm{d} x=\frac{1}{a} \sin (a x+b)+C\)

(12) \(\int \frac{1}{\cos ^{2}(a x+b)} \mathrm{d} x=\frac{1}{a} \tan (a x+b)+C\)

(13) \(\int \tan (a x+b) \mathrm{d} x=-\frac{1}{a} \ln |\cos (a x+b)|+C\)

(14) \(\int \frac{\mathrm{d} x}{a^{2}+x^{2}}=\frac{1}{a} \arctan \frac{x}{a}+C\)

(15) \(\int \frac{\mathrm{d} x}{\sqrt{x^{2}+a^{2}}}=\ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C\)

(16) \(\int \frac{d x}{x \cdot \sqrt{x^{2}-a^{2}}}=\frac{1}{a} \arccos \left|\frac{x}{a}\right|+C\)

(17) \(\int \sqrt{a^{2}-x^{2}} \mathrm{~d} x=\frac{x \sqrt{a^{2}-x^{2}}}{2}+\frac{a^{2}}{2} \arcsin \frac{x}{a}+C\)

(18) \(\int \mathrm{e}^{a x} \sin b x \mathrm{~d} x=\frac{\mathrm{e}^{a x}(a \sin b x)-b \cos b x}{a^{2}+b^{2}}+C\)

\(\begin{array}{l}I_{1}=\int \sin x d x=-\cos x+C \\ I_{2}=\int \sin (a x) d x=-\frac{1}{a} \cos (a x)+C \\ I_{3}=\int \cos x d x=\sin x+C \\ I_{4}=\int \cos (a x) d x=\frac{1}{a} \sin (a x)+C \\ I_{5}=\int \sin ^{2} x d x=\int \frac{1-\cos 2 x}{2} d x=\frac{x}{2}-\frac{\sin 2 x}{4}+C \\ I_{6}=\int \cos ^{2} x d x=\int \frac{1+\cos 2 x}{2} d x=\frac{x}{2}+\frac{\sin 2 x}{4}+C \\ I_{7}=\int \frac{d x}{\cos ^{2} x}=\tan x+C \\ I_{8}=\int \frac{d x}{\cos ^{2}(a x)}=\frac{1}{a} \tan (a x)+C \\ I_{9}=\int \frac{d x}{\sin ^{2}(a x)}=-\cot x+C \\ I_{10}=\int \frac{d x}{\sin ^{2}(a x)}=-\frac{1}{a} \cot (a x)+C \\ I_{11}=\int \tan ^{2} x d x=\int \frac{\sin x d x}{\cos x}=-\ln |\cos x|+C\end{array}\)\(\begin{array}{l}I_{12}=\int \cot x d x=\int \frac{\cos x d x}{\sin x}=\ln |\sin x|+C \\ I_{13}=\int \tan ^{2} x d x=\int\left(\frac{1}{\cos ^{2} x}-1\right) d x=\tan x-x+C \\ I_{14}=\int \cot ^{2} x d x=\int\left(\frac{1}{\sin ^{2} x}-1\right) d x=\cot x-x+C\end{array}\)