Home | | Maths 11th std | Integrals of the form e pow(ax) cos bx and e pow(ax) sin bx

Equation, Solved Example Problems | Mathematics - Integrals of the form e pow(ax) cos bx and e pow(ax) sin bx | 11th Mathematics : UNIT 11 : Integral Calculus

Chapter: 11th Mathematics : UNIT 11 : Integral Calculus

Integrals of the form e pow(ax) cos bx and e pow(ax) sin bx

The following examples illustrate that there are some integrals whose integration continues forever.

Integrals of the form


(i) e ax ax sinbx dx (ii) e ax cosbx dx

The following examples illustrate that there are some integrals whose integration continues forever. Whenever we integrate function of the form e axcos bx or eax sinbx, we have to apply the Integration by Parts rule twice to get the similar integral on both sides to solve.

Result 11.1



Caution

In applying integration by parts to specific integrals, the pair of choice for u and dv once initially assumed should be maintained for the successive integrals on the right hand side. (See the above two examples). The pair of choice should not be interchanged.


Examples 11.36

Evaluate the following integrals

(i) ∫e 3x cos 2x dx

(ii) ∫e −5x sin 3x dx

(i) ∫e 3x cos 2x dx

Using the formula



EXERCISE 11.8

Integrate the following with respect to x

(1) (i) eax cos bx (ii) e2 x sin x (iii) e x cos 2x

(2)  (i) e−3x sin 2x (ii) e−4xsin 2x (iii) e−3x cos x


Result 11.2

∫ex [ f (x) + f ′(x)]dx = ex f (x) + c

Proof

Let I = ex [ f (x) + f ′(x)]dx

= ∫ e x f (x)dx + ∫ex f ′(x)dx

Take u = f ( x); du =  f’ ( x) dx, in the first integral

dv = ex ;v = ex ,

That is, I = ex f (x) e x f ′( x)dx + ex f ′(x)dx + c

Therefore, I = e x f ( x ) + c.




EXERCISE 11.9

Integrate the following with respect to x:



Tags : Equation, Solved Example Problems | Mathematics , 11th Mathematics : UNIT 11 : Integral Calculus
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
11th Mathematics : UNIT 11 : Integral Calculus : Integrals of the form e pow(ax) cos bx and e pow(ax) sin bx | Equation, Solved Example Problems | Mathematics


Privacy Policy, Terms and Conditions, DMCA Policy and Compliant

Copyright © 2018-2023 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.