INTEGRATION OF E TO THE POWER AX PLUS B

∫e(ax+b) dx  =  (1/a)e(ax+b) + C

∫eax dx  =  (1/a) eax + C

∫ex dx  =  e+ C

Question 1 :

(i)  ∫ e3x dx

(ii)  ∫e(x+3) dx

(iii)  ∫ e(3x+2) dx

(iv)  ∫e(5-4x) dxv

(v) ∫ e-x dx

Solution :

(i)  ∫ e3x dx  =  (1/3) e3x + C

(ii)  ∫e(x+3) dx  =  (1/1) e(x+3) + C

=  (1/1) e(x+3) + C

(iii)  ∫ e(3x+2) dx  =  (1/3) e(3x+2) + C

=  e(3x+2)/3 + C 

(iv)  ∫e(5-4x) dx  =  (1/-4) e(5-4x) + C

=  -e(5-4x)/4 + C

(v) ∫ e-x dx  =  e-x/(-1) + C

= - e-x + C

(vi)  ∫ 1/e(p+qx) dx  =  ∫ e-(p+qx) dx

=  e-(p+qx)/(-q) + C

=  -1/q e(p+qx) + C

Question 2 :

(i)  ∫(ex+1)/ex dx

(ii) ∫ex log 2 ex dx

(iii)  ∫ex loga edx

Solution :

(i)  ∫(ex+1)/ex dx

=  ∫ [ (ex/ex) + 1/ex ] dx

=  ∫dx + ∫e-x dx

=  x - e-x + C

(ii) ∫ex log 2 ex dx

=  ∫e^log 2x ex dx

=  ∫2x ex dx  

=  ∫(2e)x dx

=  (2e)x/log 2e + C

(iii)  ∫ex loga edx

=  ∫e^log ax ex dx

=  ∫ax ex dx  

=  ∫(ae)x dx

=  (ae)x/log ae + C

Question 3 :

∫ 5x4 e^x5 dx

Solution :

∫ 5x4 e^x5 dx

Let u  =  x5

du  =  5x4 dx

=  ∫ eu du

=  eu + C

=  e^x5 + C

Question 4 :

∫ ex/(5+ex) dx

Solution :

∫ ex/(5+ex) dx

Let u  =  5+ex

du  =  ex dx

∫ ex/(5+ex) dx  =  ∫ du/u

=  ln u + C

=  ln (5+ex+ C

Question 5 :

∫ etan x/cos2x dx

Solution :

∫ etan x/cos2x dx

1/cos2x  =  sec2x

∫ etan x/cos2x dx  =  ∫sec2x etan x dx

Let u  =  tanx

du  =  sec2x dx

=  ∫eu dx

=  eu + C

=  etan x + C

Question 6 :

∫ e√x/√x  dx

Solution :

∫ e√x/√x  dx

Let √x  =  t

1/2√x dx  =  dt

1/√x dx  =  2 dt

=  ∫ et (2dt)

=  2∫ et dt

=  2et + C

=  2e√x + C

Question 7 :

∫ e^ sin-1 x/√(1-x2)  dx

Solution :

∫ e^ sin-1 x/√(1-x2)  dx

Let u  =  sin-1 x

du  =  1/√(1-x2) dx

∫ e^ sin-1 x/√(1-x2)  dx  =  ∫ eu du

=  eu + C

=  e^sin-1 x + C

Apart from the stuff given in this section if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. How to Solve Age Problems with Ratio

    Mar 28, 24 02:01 AM

    How to Solve Age Problems with Ratio

    Read More

  2. AP Calculus BC Integration of Rational Functions by Partical Fractions

    Mar 26, 24 11:25 PM

    AP Calculus BC Integration of Rational Functions by Partical Fractions (Part - 1)

    Read More

  3. SAT Math Practice

    Mar 26, 24 08:53 PM

    satmathquestions1.png
    SAT Math Practice - Different Topics - Concept - Formulas - Example problems with step by step explanation

    Read More