Biến đổi: ʃ\(\int\dfrac{1dx}{cosx\dfrac{\sqrt{2}}{2}\left(cosx-sinx\right)}=\int\dfrac{\sqrt{2}dx}{cos^2x\left(1-tanx\right)}=\int\dfrac{\sqrt{2}d\left(tanx\right)}{1-tanx}=-\sqrt{2}\ln trituyetdoi\left(1-tanx\right)\)
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Tính các nguyên hàm sau :
a) \(\int x\left(3-x\right)^5dx\)
b) \(\int\left(2^x-3^x\right)^2dx\)
c) \(\int x\sqrt{2-5x}dx\)
d) \(\int\dfrac{\ln\left(\cos x\right)}{\cos^2x}dx\)
e) \(\int\dfrac{x}{\sin^2x}dx\)
\(\int\dfrac{x+1}{\left(x-2\right)\left(x+3\right)}dx\)
h) \(\int\dfrac{1}{1-\sqrt{x}}dx\)
i) \(\int\sin3x\cos2xdx\)
k) \(\int\dfrac{\sin^3x}{\cos^2x}dx\)
l) \(\int\dfrac{\sin x\cos x}{\sqrt{a^2\sin^2x+b^2\cos^2x}}dx\) (\(a^2\ne b^2\))
1, \(\int\dfrac{x}{1-cos2x}dx\)
2, \(\int cos2x.e^{3x}dx\)
3, \(\int\left(2x+1\right)ln^2dx\)
4, \(\int\left(2x-1\right)cosxdx\)
5, \(\int\left(x^2+x+1\right)e^xdx\)
6, \(\int\left(2x+1\right)ln\left(x+2\right)dx\)
Tính :
\(I_1=\int\dfrac{x^2-2x+2}{\sqrt{x^2-2x}}dx\)
\(I_2=\dfrac{dx}{\left(x-1\right)\sqrt{x^2-2x+2}}\)
\(I_3=\dfrac{dx}{\left(2x+1\right)\sqrt{x^2-2x+2}}\)
Tìm nguyên hàm:
a) \(\int\left(\dfrac{1}{u^3}+\dfrac{1}{u^2}+\dfrac{1}{u}\right)du\)
b) \(\int\left(\dfrac{1}{t-2}+\dfrac{3}{1-t}\right)dt\)
c) \(\int\left(\dfrac{1}{2-3x}+\dfrac{7}{1-4x}\right)dx\)
d) \(\int e^{5x-1}dx\)
Tìm nguyên hàm của các hàm số sau:
a) \(\int\left(6x-\dfrac{1}{sin^2x}+1\right)dx\)
b) \(\int\dfrac{x^3+2x^2-1}{x^2}dx\)
1, \(\int\dfrac{lnxdx}{\sqrt{x}}\)
2, \(\int ln\left(x+\sqrt{x^2+1}\right)dx\)
3, \(\int\left(x^2+2x+3\right)dx\)
1, \(\int\dfrac{x^3dx}{\left(x^8-4\right)^2}\)
2, \(\int\dfrac{2x+1}{x^4+2x^2+3x^2+2x-3}dx\)
3, \(\int\dfrac{sin\sqrt[3]{x}}{\sqrt[3]{x^2}}dx\)
4, \(\int\dfrac{dx}{sin^2x+2cos^2x}\)
5, \(\int\dfrac{sinx+cosx}{3+sin2x}dx\)
Áp dụng phương pháp tính nguyên hàm từng phần, hãy tính :
a) \(\int\left(1-2x\right)e^xdx\)
b) \(\int xe^{-x}dx\)
c) \(\int x\ln\left(1-x\right)dx\)
d) \(\int x\sin^2xdx\)
e) \(\int\ln\left(x+\sqrt{1+x^2}\right)dx\)
g) \(\int\sqrt{x}\ln^2xdx\)
h) \(\int x\ln\dfrac{1+x}{1-x}dx\)
Tính:
\(A=\int\dfrac{cosx}{\left(x+1\right)^2}dx\)