\(I=\dfrac{1}{2}\int f\left(x^2\right).d\left(x^2\right)=\dfrac{1}{2}x^2\sqrt{\left(x^2\right)^2+1}=\dfrac{1}{2}x^2\sqrt{x^4+1}\)
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Bài 1: Nguyên hàm
Các câu hỏi tương tự
Cho \(\int f\left(x\right)dx=x\sqrt{x^2+1}.\: \)Tìm \(I=\int x.f\left(x^2\right)dx\)
Giải giúp em với, em cảm ơn
1) \(\int\left(\frac{lnx}{2+lnx}\right)^2\)
2) \(\int\frac{dx}{\left(x+3\right)^3\left(x+5\right)^5}\)
3) \(\int\frac{xdx}{\sqrt{1+\sqrt[3]{x^2}}}\)
4) \(\int\frac{dx}{x^3.\sqrt[3]{2-x^3}}\)
5)\(\int\sqrt[3]{\frac{2-x}{2+x}}.\frac{1}{\left(2-x\right)^2}dx\)
Tính các nguyên hàm sau :
a) int xleft(3-xright)^5dx
b) intleft(2^x-3^xright)^2dx
c) int xsqrt{2-5x}dx
d) intdfrac{lnleft(cos xright)}{cos^2x}dx
e) intdfrac{x}{sin^2x}dx
intdfrac{x+1}{left(x-2right)left(x+3right)}dx
h) intdfrac{1}{1-sqrt{x}}dx
i) intsin3xcos2xdx
k) intdfrac{sin^3x}{cos^2x}dx
l) intdfrac{sin xcos x}{sqrt{a^2sin^2x+b^2cos^2x}}dx (a^2ne b^2)
Đọc tiếp
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, \(\int\dfrac{lnxdx}{\sqrt{x}}\)
2, \(\int ln\left(x+\sqrt{x^2+1}\right)dx\)
3, \(\int\left(x^2+2x+3\right)dx\)
Tính :
\(I_1=\int\dfrac{x^2-2x+2}{\sqrt{x^2-2x}}dx\)
\(I_2=\dfrac{dx}{\left(x-1\right)\sqrt{x^2-2x+2}}\)
\(I_3=\dfrac{dx}{\left(2x+1\right)\sqrt{x^2-2x+2}}\)
1)intsqrt{frac{1-sqrt{x}}{1+sqrt{x}}}dx
2)intfrac{dx}{left(e^x+1right)left(x^2+1right)}
3)intfrac{1+2xsqrt{1-x^2}+2x^2}{1+x+sqrt{1+x^2}}dx
4)intfrac{sin^6x+ctext{os}^6x}{1+6^x}dx
5)int_0^{frac{pi}{2}}frac{sqrt{ctext{os}x}}{sqrt{stext{inx}}+sqrt{ctext{os}x}}dx
6)intfrac{x^4}{2^x+1}dx
7)int_0^{frac{pi^2}{4}}sinsqrt{x}dx
8)intsqrt[6]{1-ctext{os}^3x}.stext{inx}.ctext{os}^5xdx
9)intsqrt{frac{1}{4x}+frac{sqrt{x}+e^x}{sqrt{x}.e^x}}dx
10)intfrac{ctext{os}x+stext{inx}}{left(e^xstext{inx}+1right)...
Đọc tiếp
1)\(\int\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}dx\)
2)\(\int\frac{dx}{\left(e^x+1\right)\left(x^2+1\right)}\)
3)\(\int\frac{1+2x\sqrt{1-x^2}+2x^2}{1+x+\sqrt{1+x^2}}\)dx
4)\(\int\frac{sin^6x+c\text{os}^6x}{1+6^x}dx\)
5)\(\int_0^{\frac{\pi}{2}}\frac{\sqrt{c\text{os}x}}{\sqrt{s\text{inx}}+\sqrt{c\text{os}x}}dx\)
6)\(\int\frac{x^4}{2^x+1}dx\)
7)\(\int_0^{\frac{\pi^2}{4}}sin\sqrt{x}dx\)
8)\(\int\sqrt[6]{1-c\text{os}^3x}.s\text{inx}.c\text{os}^5xdx\)
9)\(\int\sqrt{\frac{1}{4x}+\frac{\sqrt{x}+e^x}{\sqrt{x}.e^x}}dx\)
10)\(\int\frac{c\text{os}x+s\text{inx}}{\left(e^xs\text{inx}+1\right)s\text{inx}}dx\)
Áp dụng phương pháp tính nguyên hàm từng phần, hãy tính :
a) intleft(1-2xright)e^xdx
b) int xe^{-x}dx
c) int xlnleft(1-xright)dx
d) int xsin^2xdx
e) intlnleft(x+sqrt{1+x^2}right)dx
g) intsqrt{x}ln^2xdx
h) int xlndfrac{1+x}{1-x}dx
Đọc tiếp
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) \(\int x\ln\dfrac{1+x}{1-x}dx\)
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, \(\int\left(2x+1\right)ln\left(x+2\right)dx\)
\(\int\dfrac{1}{cosx.cos\left(x+\dfrac{\pi}{4}\right)}dx\)
\(\int\dfrac{1}{x^3\left(1+x^2\right)}dx=\dfrac{a}{x^2}+blnx+cln\left(1+x^2\right).S=a+b+c=?\)
\(\int\dfrac{5-3x}{\left(x^2-5x+6\right)\left(x^2-2x+1\right)}dx=\dfrac{a}{x-1}-ln\left(\dfrac{x-b}{x-c}\right)+C.P=2a+b\)