Integrals of Exponential Functions

Key Questions

• What is the antiderivative of #n^x#?

  By rewriting a bit,

  #int n^x dx=int e^{(lnn)x}dx=e^{(lnn)x}/{lnn}+C=n^x/{lnn}+C#

  I hope that this was helpful.

  Wataru · 3 · Oct 16 2014
• How do you evaluate the integral #inte^(-x) dx#?

  The answer is #I=-e^(-x)+C#.

  This integral can be solved by a substitution:

  #u=-x#
  #du=-dx#
  #-du=dx#

  So, now we can substitute:

  #int e^(-x)dx = int e^u (-du)#
  #=-int e^u du#
  #=-e^u + C#

  and substitute back:
  #=-e^(-x) + C#

  For simple looking integrands, you should try a quick check to see if substitution works before trying harder integration methods.

  Amory W. · · Aug 20 2014
• How do you evaluate the integral #int3^(x) dx#?

  Answer:

  #int3^xdx=1/ln3 3^x+"c"#

  Explanation:

  We want to find #int3^xdx#.

  Make the natural substitution #u=3^x# so #du=3^xln3dx#.

  So

  #int3^xdx=1/ln3int1du=1/ln3 u+"c"=1/ln3 3^x+"c"#

  Monzur R. · 1 · Mar 14 2018
• What is the antiderivative of #e^x#?

  Since

  #(e^x)'=e^x#,

  we have

  #int e^x dx=e^x+C#.

  I hope that this was helpful.

  Wataru · 4 · Oct 16 2014

• How do you evaluate the integral #inte^(4x) dx#?
• How do you evaluate the integral #inte^(-x) dx#?
• How do you evaluate the integral #int3^(x) dx#?
• How do you evaluate the integral #int3e^(x)-5e^(2x) dx#?
• How do you evaluate the integral #int10^(-x) dx#?
• What is the integral of #e^(x^3)#?
• What is the integral of #e^(0.5x)#?
• What is the integral of #e^(2x)#?
• What is the integral of #e^(7x)#?
• What is the integral of #2e^(2x)#?
• How do I find the antiderivative of #f(x)=1/(e^(2x)-9)^(1/2)#?
• What is the antiderivative of #ln(x)#?
• How do I find the antiderivative of #f(x)=e^(-2x)#?
• How do I find the antiderivative of #f(x)=e^(-5x)#?
• How do you calculate the double integral of #f(x,y) = 28y(e^x)# over the triangle indicated by the following points (0,0), (4,1), and (4,3)?
• What is the antiderivative of #pi^(x-1)dx#?
• How do you evaluate the integral #e^(sqrt x)dx# when #a=1# and #b=9#?
• How do you integrate #e^(x^2)#?
• How do you find the antiderivative of #dx/(e^(2x)-9)^(1/2)#?
• What is the antiderivative of #-ln|cscx+cotx|#?
• How do you integrate #int5e^xdx#?
• How do you find the antiderivative of #e^(-2x)#?
• What is the antiderivative of #e^(2x)#?
• What is the integral of #(e^(2x))/(1+e^(2x))#?
• What is the integral of #e^(2x)#?
• What is anti derivative of lnx?
• What is the antiderivative of ln(x)/x?
• How do you find the integration of #log x#?
• How do you integrate #(1 + e^x )/(1 - e^x)#?
• How do you integrate #e^-lnx#?
• What is the integral of #ln(3x)#?
• What is the integral of #ln(2x)#?
• How do you find the integral of #log_8 (2x+1) dx#?
• What is the integral of #ln(sqrt(x))#?
• How do you find the integral of #e^(x^2)#?
• How do you find the integral of #e^(-x)dx#?
• What is the integral of #e^(-x^2)#?
• What is the integral of #e^(1/x)dx#?
• What is the integral of #e^(2x^2)#?
• What is the integral of #dy/e^y#?
• What is the integral of #e^(-0.2t) dt#?
• What is the integral of #e^(3x)#?
• How do you integrate #10^x dx# from 1 to 0?
• How do you find the integral of #ln(1+x^2)#?
• How do you integrate #(lnx)^2#?
• What is the integral of #[ln(lnx)]/[x] dx#?
• How do you integrate #(lnx)/(2x)#?
• How do you find the integral of #(x^3)(lnx) dx#?
• What is the integral of #1/ln(lnx)#?
• How do you integrate #intdx/(x+xlnx^2)#?
• What is the antiderivative of #e^(-3x)#?
• What is the antiderivative of #3/x#?
• What is the antiderivative of #e^-x#?
• What is the antiderivative of #-5e^(x-1)#?
• What is the antiderivative of #(e^x)/(1+e^2x)#?
• What is the antiderivative of #e^(x^2)#?
• What is the antiderivative of # 1/(xlnx)#?
• How do you integrate ln(x+1)?
• What does it mean to Fourier transform a function, i.e. #hat(f)(omega) = mathcalF_(omega)[f(x)]#?
• How do you find the integral of ##?
• How do you find the integral of #(1 + e^2x) ^(1/2)#?
• How do you find the integral of #e^((-1/2)*x)#?
• What is #int (lnx)^2 / x^3#?
• What is #int lnx^2 / lnx^3#?
• What is #int ((x+1)*ln(x+1))/(x+2)-x#?
• What is #int (lnx) / x^(1/2)#?
• What is #int_0^pi (lnx)^2 / x^(1/2)#?
• What is #int_0^pi (lnx)^2#?
• What is #int_0^pi sin^2(lnx)#?
• What is #int_(pi/2)^pi lnsinx#?
• What is #int_(pi/2)^pi x^2ln(tanx)#?
• What is #int_1^ln5 xe^(x^2)+x^2e^x+x^3+e^(x^3) dx#?
• What is #int(lnx^3)/x^4 dx#?
• What is #int(lnx^9)/x^4 dx#?
• What is #int ln(lnx)/xdx#?
• What is #int 1/(xlnx)dx#?
• What is #int (1+lnx)/x^2dx#?
• What is #int ((1+lnx)/x)^2dx#?
• What is #int x^2lnxdx#?
• What is #int (ln(sinx))*cosxdx#?
• What is the antiderivative of the natural logarithm of x?
• What is the antiderivative of #ln(x)/x#?
• What is the antiderivative of #ln(x)^2#?
• What is the antiderivative of #ln(x+1)#?
• What is the antiderivative of # ln(2x)#?
• What is the antiderivative of #xlnx#?
• What is the antiderivative of # (ln x)^2/x^2#?
• What is the antiderivative of #[ln(ln x)]/x#?
• What is the antiderivative of #e^(x^2)#?
• What is the antiderivative of #ln(x)^2#?
• What is the antiderivative of #ln(2x)/x^(1/2)#?
• How do you integrate #lnx / x#?
• Question #8f777
• How do you find the antiderivative of #e^(2x) dx#?
• How do you find the antiderivative of #e^-x#?
• How do you find the antiderivative of #e^(2x) * sqrt(e^x + 1)#?
• How do you find the antiderivative of #(e^(2x))/(1+e^(2x))#?
• How do you find the antiderivative of #e^(-2x^2)#?
• How do you find the antiderivative of #x^2 e^(2x)#?
• How do you find the antiderivative of #e^(2x)*sin(3x) dx#?
• How do you find the antiderivative of #e^(2x) / (3+e^(2x))#?
• How do you find the antiderivative of #e^(2x)*cos[3x] dx#?
• How do you find the antiderivative of #[e^(2x)/(4+e^(4x))]#?
• How do you find the antiderivative of # {(e^x)/ [(e^(2x)) - 1]}#?
• How do you find the antiderivative of #(e^(2x)+e^x)/(e^(2x)+1)#?
• How do you find the antiderivative of #(e^-2x)^2#?
• How do you find the antiderivative of #e^(3x)#?
• How do you find the antiderivative of #e^(2x)/sqrt(1-e^x)#?
• How do you find the antiderivative of #(x^2)(e^x)dx#?
• How do you find the antiderivative of #e^(2x)*( tan(e^(2x)) )^2#?
• How do you find the antiderivative of #(e^(2x))sin3x#?
• How do you find the antiderivative of #e^(2x)*sin(e^x)dx#?
• How do you find the antiderivative of #e^(2x)(sin x)#?
• How do you find the antiderivative of #((2x)e^(3x))#?
• How do you find the antiderivative of #e^x(1-(e^-x)sec^2x) dx#?
• How do you find the antiderivative of #e^(-2x)#?
• How do you find the antiderivative of # (e^(2x))(sqrt(1+3e^(2x)))#?
• How do you find the antiderivative of #(e^x)/(1+e^(2x))#?
• How do you find the antiderivative of # [(e^(2x)) +4]^2 dx#?
• How do you find the antiderivative of #(1 + e^(2x)) ^(1/2)#?
• How do you find the antiderivative of #x^2*e^(2x)#?
• How do you find the antiderivative of #(e^(2x))/(1+(e^(4x))dx#?
• How do you find the antiderivative of #e^(2x) * sin(4x)dx#?
• How do you find the antiderivative of #(cos 2x)e^(sin2x)#?
• What is the integral of #xln^2(x+3)#?
• How do you integrate #(e^(x-2))^2/e^(2x)#?
• How do you integrate # (3-x)e^(-2x)#?
• How do you integrate #f(x) = e^-|13x|#?
• How do you integrate #e^(sinx) cosx dx#?
• How do you integrate #f(t) = 1.4e^(0.07t)#?
• How do you integrate #y=xe^(-x)#?
• How do you integrate #(e^(sqrt(1+3x)))dx#?
• How do you integrate # 3^x#?
• How do you integrate # (e^(2x-1))-1#?
• How do you integrate # (x-1) e^(-x^2+2x) dx#?
• How do you integrate # (1/(e^x+1))dx #?
• How do you integrate #(e^x/x)dx #?
• How do you integrate #e^x / sqrt(1-e^(2x)) dx#?
• How do you integrate #sin(x)*(e)^(2 x) dx#?
• How do you integrate #(x^2e^(x/2))dx#?
• How do you integrate #e^x/e^(2x+1)#?
• How do you integrate #e^sqrt(z) /sqrt(z) dz#?
• How do you integrate #(t^2)e^(4t)#?
• How do you integrate #x5^(x^2) dx#?
• How do you integrate #int e^(5x)(5)dx#?
• How do you integrate #int e^sqrtx/sqrtxdx#?
• How do you integrate #int e^(1/x^2)/x^3dx#?
• How do you integrate #int e^-x/(1+e^-x)dx#?
• How do you integrate #int e^(2x)/(1+e^(2x))dx#?
• How do you integrate #int e^xsqrt(1-e^x)dx#?
• How do you integrate #int (e^x-e^-x)/(e^x+e^-x)dx#?
• How do you integrate #int (e^x+e^-x)/(e^x-e^-x)dx#?
• How do you integrate #int (5-e^x)/(e^(2x))dx#?
• How do you integrate #int (e^(2x)+2e^x+1)/(e^x)dx#?
• How do you integrate #int e^-xtan(e^-x)dx#?
• How do you integrate #int ln(e^(2x-1))dx#?
• How do you integrate #int e^(-2x)dx# from #[0,1]#?
• How do you integrate #int e^(3-x)dx# from #[3,4]#?
• How do you integrate #int xe^(-x^2)dx# from #[0,1]#?
• How do you integrate #int x^2e^(x^3/2)dx# from #[-2,3]#?
• How do you integrate #int e^(3/x)/x^2dx# from #[0,1]#?
• How do you integrate #int xe^(-x^2/2)# from #[0,sqrt2]#?
• How do you integrate #int e^(sinpix)cospix# from #[0,pi/2]#?
• How do you integrate #int e^(sec2x)sec2xtan2xdx# from #[pi/3,pi/2]#?
• How do you integrate #int 3^xdx#?
• How do you integrate #int 5^-xdx#?
• How do you integrate #int x(5^(-x^2))dx#?
• How do you integrate #int (3-x)7^((3-x) ^2)dx#?
• How do you integrate #int 2^sinxcosxdx#?
• How do you integrate #int 2^xdx# from #[-1,2]#?
• How do you integrate #int 4^(x/2)dx# from #[-2,2]#?
• How do you integrate #int 5^x-3^xdx# from #[0,1]#?
• How do you integrate #int 6^x-2^xdx# from #[1,e]#?
• How do you find the indefinite integral of #int 3^x#?
• How do you find the indefinite integral of #int 5^-x#?
• How do you find the indefinite integral of #int x(5^(-x^2))#?
• How do you find the indefinite integral of #int (3-x)7^((3-x)^2)#?
• How do you find the indefinite integral of #int (3^(2x))/(1+3^(2x))#?
• How do you find the indefinite integral of #int 2^(sinx)cosx#?
• Question #bd150
• #lim_(xrarr0)(e^(2x)+x)^(1//x)=# ?
• How do you evaluate the integral #int dx/e^(3x)#?
• How do you evaluate the integral #int xe^(-x^2)#?
• How do you evaluate the integral #int e^(2x)sqrt(1+e^(2x))#?
• How do you evaluate the integral #int x^2e^x#?
• How do you evaluate the integral #int e^(5x)#?
• How do you evaluate the integral #int e^-xcosx#?
• Question #2ce29
• Evaluate the following, #int_0^1 (x^e +e^x) dx#. I know the anti-derivative of x^e is #x^(e+1)/(e+1)#. How would i solve this,would i use F(b) +F(a)?
• Question #3ba38
• Integral of #(x^23)/(x^32+1)#?
• How do you integrate #int_ln2^ln3(e^x/sqrt(e^(2x)+1)dx# ?
• Evaluate #int(9e^(9t))/(e^(18t)+11e^(9t)+18)dt#?