Title: integrales inmediatas

Q: [$]\displaystyle\int\frac{2}{x+3}\,\mbox{d}x[/$]
A: [$]2\ln|x+3|+C[/$]

Q: [$]\displaystyle\int\frac{3}{(x+1)^2}\,\mbox{d}x[/$]
A: [$]\displaystyle-\frac{3}{x+1}+C[/$]

Q: [$]\displaystyle\int\frac{2}{x^2+3}\,\mbox{d}x[/$]
A: [$]\frac {2}{\sqrt{3}}\arctan\frac{x}{\sqrt{3}}+C[/$]

Q: [$]\displaystyle\int\frac{1}{1+4x^2}\,\mbox{d}x[/$]
A: [$]\frac{1}{2}\,\arctan2x+C[/$]

Q: [$]\displaystyle\int\frac{x}{1+x^2}\,\mbox{d}x[/$]
A: [$]\frac{1}{2}\ln(1+x^2)+C[/$]

Q: [$]\displaystyle\int\frac{3}{1+2x^2}\,\mbox{d}x[/$]
A: [$]\frac{3}{\sqrt{2}}\arctan(\sqrt{2}x)+C[/$]

Q: [$]\displaystyle\int\,\frac{3x}{4+x^2}\,\mbox{d}x=\frac{3}{2}\int\frac{2x}{4+x^2}\,\mbox{d}x[/$]
A: [$]\frac{3}{2}\ln(4+x^2)+C[/$]

Q: [$]\displaystyle\int\cos x\,\mbox{d}x[/$]
A: [$]\sin x+C[/$]

Q: [$]\displaystyle\int\cos2x\,\mbox{d}x[/$]
A: [$]\frac{1}{2}\sin2x+C[/$]

Q: [$]\displaystyle\int\cos(x+\pi)\,\mbox{d}x[/$]
A: [$]\sin(x+\pi)+C=-\sin x+C[/$]

Q: [$]\displaystyle\int\cos(\pi-x)\,\mbox{d}x[/$]
A: [$]-\sin(\pi-x)+C=-\sin x+C[/$]

Q: [$]\displaystyle\int2\cos3x\,\mbox{d}x[/$]
A: [$]\frac{2}{3}\sin3x+C[/$]

Q: [$]\displaystyle\int\sin x\,\mbox{d}x[/$]
A: [$]-\cos x+C[/$]

Q: [$]\displaystyle\int\tan x\,\mbox{d}x[/$]
A: [$]-\ln|\cos x|+C=\ln|\sec x|+C[/$]

Q: [$]\displaystyle\int\cot x\,\mbox{d}x[/$]
A: [$]\ln|\sin x|+C[/$]

Q: [$]\displaystyle\int\frac{1}{\sqrt{1-x^2}}\,\mbox{d}x[/$]
A: [$]\arcsin x+C[/$]

Q: [$]\displaystyle\int\frac{1}{\sqrt{4-x^2}}\,\mbox{d}x[/$]
A: [$]\arcsin\frac{x}{2}+C[/$]

Q: [$]\displaystyle\int x\cos x\,\mbox{d}x=\int x\,\mbox{d}(\sin x)[/$]
A: [$]x\sin x+\cos x+C[/$]

Q: [$]\displaystyle\int\frac{1}{\sqrt{x}}\,\mbox{d}x=\int\frac{2}{2\sqrt{x}}\,\mbox{d}x[/$]
A: [$]2\sqrt{x}+C[/$]

Q: [$]\displaystyle\int\frac{3}{\sqrt{1-4x^2}}\,\mbox{d}x[/$]
A: [$]\frac{3}{2}\arcsin2x+C[/$]

Q: [$]\displaystyle\int x\sin x\,\mbox{d}x=\int x\,\mbox{d}(-\cos x)[/$]
A: [$]-x\cos x+\sin x+C[/$]

Q: [$]\displaystyle\int\frac{1}{2\sqrt{x}}\,\mbox{d}x[/$]
A: [$]\sqrt{x}+C[/$]

Q: [$]\displaystyle\int\frac{1}{\sqrt{x}}\,\mbox{d}x[/$]
A: [$]2\,\sqrt{x}+C[/$]

Q: [$]\displaystyle\int xe^x\,\mbox{d}x=\int x\,\mbox{d}(e^x)[/$]
A: [$]xe^x-e^x+C[/$]

Q: [$]\displaystyle\int\ln x\,\mbox{d}x[/$]
A: [$]x\ln x-x+C[/$]

Q: [$]\displaystyle\int e^x\,\mbox{d}x[/$]
A: [$]e^x+C[/$]

Q: [$]\displaystyle\int e^{2x}\,\mbox{d}x[/$]
A: [$]\frac{1}{2}\,e^{2x}+C[/$]

Q: [$]\displaystyle\int e^{x+5}\,\mbox{d}x[/$]
A: [$]e^{x+5}+C[/$]

Q: [$]\displaystyle\int e^{\frac{x}{2}}\,\mbox{d}x[/$]
A: [$]2e^\frac{x}{2}+C[/$]

Q: [$]\displaystyle\int(x+2)^4 \,\mbox{d}x[/$]
A: [$]\displaystyle\frac{(x+2)^5}{5}+C[/$]

Q: [$]\displaystyle\int(2x-1)^3 \,\mbox{d}x[/$]
A: [$]\dfrac{1}{2}\,\dfrac{(2x+1)^4}{4}+C[/$]

Q: [$]\displaystyle\int (x+3)^2\,\mbox{d}x[/$]
A: [$]\dfrac{(x+3)^3}{3}+C[/$]

Q: [$]\displaystyle\int\frac{1}{x^2} \,\mbox{d}x[/$]
A: [$]\displaystyle-\frac{1}{x}+C[/$]

Q: [$]\displaystyle\int\frac{2}{(x+3)^2} \,\mbox{d}x[/$]
A: [$]\displaystyle-\frac{2}{x+3}+C[/$]

Q: [$]\displaystyle\int\frac{2x+1}{x^2+x-2} \,\mbox{d}x[/$]
A: [$]\ln|x^2+x-2|+C[/$]

Q: [$]\displaystyle\int \frac{x+3}{x^2+6x-1}\,\mbox{d}x[/$]
A: [$]\frac{1}{2}\,\ln|x^2+6x-1|+C[/$]

Q: [$]\displaystyle\int\frac{1}{x^2+4x+7} \,\mbox{d}x=\int\frac{1}{(x+2)^2+3}\,\mbox{d}x[/$]
A: [$]\frac{1}{\sqrt{3}}\,\arctan\frac{x+2}{\sqrt{3}}+C[/$]

Q: [$]\displaystyle\int \frac{1}{x^2-2x+5}\,\mbox{d}x=
\int\frac{1}{(x-1)^2+4}\,\mbox{d}x[/$]
A: [$]\frac{1}{2}\,\arctan\frac{x-1}{2}+C[/$]

Q: [$]\displaystyle\int \frac{3x+1}{x^2-2x-3}\,\mbox{d}x
=\int\frac{A}{x-3}\,\mbox{d}x+\int\frac{B}{x+1}\,\mbox{d}x[/$]
A: [$]\frac{5}{2}\,\ln|x-3|+\frac{1}{2}\,\ln|x+1|+C[/$]

Q: [$]\displaystyle\int\frac{x-3}{x^2+4x+4}\,\mbox{d}x
=\int\frac{A}{x+2}\,\mbox{d}x
+\int\frac{B}{(x+2)^2}\,\mbox{d}x[/$]
A: [$]\ln|x+2|+\dfrac{5}{x+2}+C[/$]

Q: [$]\displaystyle\int\sin^2x\,\mbox{d}x=\frac{1}{2}\int(1-\cos2x)\,\mbox{d}x[/$]
A: [$]\frac{1}{2}\,x-\frac{1}{4}\,\sin2x+C[/$]

Q: [$]\displaystyle\int\cos^2x\,\mbox{d}x=\frac{1}{2}\int(1+\cos2x)\,\mbox{d}x[/$]
A: [$]\frac{1}{2}\,x+\frac{1}{4}\,\sin2x+C[/$]

Q: [$]\displaystyle\int\frac{4x+3}{x^2-2x+5} \,\mbox{d}x
=A\int \frac{2x-2}{x^2-2x+5}\,\mbox{d}x
+B\int \frac{1}{(x-1)^2+4}\,\mbox{d}x[/$]
A: [$]2\ln(x^2-2x+5)+\frac{7}{2}\,\arctan\frac{x-1}{2}+C[/$]

Q: [$]\displaystyle\int\sqrt{x} \,\mbox{d}x[/$]
A: [$]\dfrac{x^{\frac{3}{2}}}{\frac{3}{2}}+C=\dfrac{2\sqrt{x^3}}{3}+C[/$]

Q: [$]\displaystyle\int\sqrt{x+1}\, \mbox{d}x[/$]
A: [$]\dfrac{(x+1)^{\frac{3}{2}}}{\frac{3}{2}}+C=\dfrac{2\sqrt{(x+1)^3}}{3}+C[/$]

Q: [$]\displaystyle\int\sqrt{3x+1}\, \mbox{d}x[/$]
A: [$]\dfrac{1}{3}\dfrac{(3x+1)^{\frac{3}{2}}}{\frac{3}{2}}+C=\dfrac{2\sqrt{(3x+1)^3}}{9}+C[/$]

Q: [$]\displaystyle\int \frac{\sin x}{1+\cos x}\,\mbox{d}x[/$]
A: [$]-\ln|1+\cos x|+C[/$]

Q: [$]\displaystyle\int \frac{e^x}{e^x+1}\,\mbox{d}x[/$]
A: [$]\ln|e^x+1|+C[/$]

Q: [$]\displaystyle\int \frac{\cos x-\sin x}{\cos x+\sin x}\,\mbox{d}x[/$]
A: [$]\ln|\cos x+\sin x|+C[/$]

Q: [$]\displaystyle\int \frac{x+1}{x-1}\,\mbox{d}x=\int\left(1+ \frac{2}{x-1}\right)\,\mbox{d}x[/$]
A: [$]x+2\ln|x-1|+C[/$]

Q: [$]\displaystyle\int e^{\sin x}\cos x \,\mbox{d}x=\int e^{\sin x}\,\mbox{d}(\sin x)[/$]
A: [$]e^{\sin x}+C[/$]

Q: [$]\displaystyle\int\frac{1}{\cos^2x}\,\mbox{d}x=\int\sec^2x\,\mbox{d}x[/$]
A: [$]\tan x+C[/$]

Q: [$]\displaystyle\int\frac{1}{\sin^2x}\,\mbox{d}x=\int\csc^2x\,\mbox{d}x[/$]
A: [$]-\cot x+C[/$]