Question

answer all

Find the indefinite integral. (Use C for the constant of integration.) el/t9 too

Answer 1

Answer 2

SOLUTION :

1.

∫ e^(1/x^9) / x^10 dx 

= ∫ e^(x^(-9)) x^(-10) dx

Let x^(-9) = u

Differentiating

=> - 9 x^(-10) dx = du

=> x^(-10) dx = - 1/9 du

So.

Given integral 

=  - 1/9 ∫ e^u du 

= - 1/9 e^u + C

= - 1/9 e^(x^(-9)) + C

= - 1/9 e^(1/x^9) + C (ANSWER)

2.

Using t for theta.

∫ t tan (9 t^2) dt 

Let 9 t^2 = u

Differentiating :

=> 18 t dt = du

=> t dt = du/18

So,

Given integral

= 1/18 ∫ tan(u) du 

= 118 ∫ sin(u) / cos(u) du

Let cos(u) = v

Differentiating :

- sin(u) du = dv

So.

Given integral 

= - 1/18 ∫ dv / v

= - 1/18 ln(v) + C

= - 1/18 ln(cos(u)) + C

= - 1/18 ln(cos(9 t^2)) + C (ANSWER)

3.

∫ 3 / (16 + (4 - x)^2)  dx

Let 

(4 - x) = 4 u 

Differentiating :

- dx = 4 du 

=> dx = - 4 du

16 + (4 - x)^2 = 16 + (4u)^2 = 16(1 + u^2)

So,

Given integral

= - 4/16 * 3 ∫ du / (1 + u^2) 

= - 3/4 tan^(-1) (u) + C

= - 3/4 tan^(-1) ((4 - x) / 4) + C (ANSWER).