Integrate the function f(x)=3^x+2 with respect to x

First note that a=eln(a) and ln(ab)=bln(a)By substituting a=3x we get a=3x=eln(3^x)=exln(3), and hence f(x)=exln(3)+2∫ f(x)dx=∫ exln(3)+2 dx . First we can split this into the sum of two integrals ∫ exln(3)dx + ∫ 2 dxRemember that d/dx(eg(x)) for some function g is equal to g'(x)eg(x) by the chain rule so ∫ exln(3)dx must equal 1/ln(3)exln(3) as 1/ln(3)d/dx(xln(3))=1/ln(3)ln(3)=1And ∫ 2 dx is solved by simply raising the power of any x elements by 1 and dividing the coefficient by this raised power. Hence ∫ 2 dx=2/1x0+1=2x=2xSo ∫ f(x)dx=(1/ln(3)exln(3))+2x+c (remembering the constant) and by the previous substitution 3x=ex*ln(3) ∫ f(x)dx=1/ln(3)*3x+2x+c

CC
Answered by Christian C. Maths tutor

2841 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Solve the quadratic inequality: x^2 - 5x + 4 < 0


if f(x) = 4x^2 - 16ln(x-1) - 10, find f'(x) and hence solve the equation f'(x)=0.


What is De Moivre's theorem?


Integrate ((7e^(x/2))/4) with respect to x within the bounds of x=0 and x=2. (Basic introduction to definite integration)


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences