In this page integration worksheet5 solution5 we are going to see
solution of some practice question from the worksheet of integration.

**Question 15**

Integrate the following with respect to x, sin (log x)/x

**Solution:**

We are going to solve this problem by using substitution method. For that we are going to consider log x as "t"

t = log x

differentiating with respect to "x"

dt = (1/x) dx

= ∫ sin (log x)/x dx

= ∫ sin t dt

= cos t + C

= cos (log x) + C

**Question 16**

Integrate the following with respect to x, cot x/log sin x

**Solution:**

We are going to solve this problem by using substitution method. For that we are going to consider log x as "t"

t = log sin x

dt =(1/sin x)cos x dx

dt = cot x

= ∫ cot x/log sin x dx

= ∫ (1/t) dx

= log t + C

= log (log sin x) + C

**Question 17**

Integrate the following with respect to x, sec⁴xtan x

**Solution:**

= ∫ sec⁴xtan x dx

= ∫ sec ³x (sec x tan x) dx

t = sec x

differentiating with respect to "x"

dt = sec x tan x dx

= ∫ t³ dt

= t^(3+1)/(3+1) + C

= (t⁴/4) + C

= (sec ⁴ x/4) + C

**Question 18**

Integrate the following with respect to x, tan³x sec x

**Solution:**

= ∫ tan³x sec x dx

t = sec x

differentiating with respect to "x"

dt = sec x tan x dx

= ∫ tan ²x tan x sec x dx

= ∫ (sec ²x - 1) tan x sec x dx

= ∫ (sec³ x tan x - tan x sec x) dx

= ∫ sec² x sec x tan x dx - ∫ tan x sec x dx

= ∫ t² dt - ∫ dt

= t^(2+1)/(2+1) - t + C

= (t⁴/4) + C

= (sec ⁴ x/4) + C

**Question 19**

Integrate the following with respect to x, sin x/sin (x + a)

**Solution:**

= ∫ sin x/sin (x + a) dx

= ∫ sin (x + a - a)/sin (x + a) dx

now we are going to compare the numerator with the formula sin (A- B) = sin A cos B - cos A sin B

= ∫ [sin (x + a) cos a - cos (x + a) sin a]/sin (x + a) dx

= ∫ [sin (x + a) cos a]/sin (x + a) dx - ∫[cos (x + a) sin a]/sin (x + a) dx

= ∫ cos a dx - ∫cot (x + a) sin a dx

= cos a x - sin a log sin (x + a) + C

integration worksheet5 solution5 integration worksheet5 solution5

- Back to worksheet
- Integration
- Substitution method
- Decomposition method
- Properties of integrals
- Integration-by parts
- Integration-of Sec³ x
- Standard integrals
- Integrating quadratic denominator
- Integration-using partial fractions
- Definite integrals