In this page integration worksheet2 solution10 we are going to see
solution of some practice question from the worksheet of integration.

**Question 10**

(i) Integrate the following with respect to x , tan (3 - 4 x)/cos (3 - 4 x)

**Solution:**

∫ tan (3 - 4 x)/cos (3 - 4 x) dx

Let us split tan (3 - 4 x)/cos (3 - 4 x) as tan (3 - 4 x) **x** [1/cos(3 - 4 x)]

∫ tan (3 - 4 x)/cos (3 - 4 x) dx = ∫tan (3 - 4 x) sec (3 - 4 x) dx

The given question exactly matches the formula

**∫ sec (a x + b) tan (a x + b) = (1/a) sec (a x + b) + C**

Now we are going to integrate the given question by using this formula in the question instead of "a" we have -4

So we get,

∫tan (3 - 4 x) sec (3 - 4 x) dx = -(1/-4) sec (3 - 4 x) + C

= -sec (3 - 4 x)/4 + C

(ii) Integrate the following with respect to x , 1/e^(p + q x)

∫ 1/e^(p + q x) dx = ∫ e^-(p + q x) dx

= e^-(p + q x)/-q + C

= -1/q e^(p + q x) + C

(iii) Integrate the following with respect to x ,1/tan (2 x + 3) sin (2 x + 3)

**Solution:**

∫ 1/tan (2 x + 3) sin (2 x + 3) dx

Let us split 1/tan (2 x+3) sin (2 x+3) as 1/tan (2 x+3) **x** 1/sin (2 x+3)

Now we are going to write 1/tan (2 x+3) as cot (2 x + 3) and 1/sin(2x+3) as cosec (2x + 3)

∫ 1/tan (2 x + 3) sin (2 x + 3) dx = ∫cot (2 x + 3) cosec (2 x + 3) dx

The given question exactly matches the formula

**∫ cosec (a x + b) cot (a x + b) = -(1/a) cosec (a x + b) + C**

Now we are going to integrate the given question by using this formula in the question instead of "a" we have 2

So we get,

∫cot (2 x + 3) cosec (2 x + 3) dx = -(1/2) cosec (2 x + 3) + C

= -cosec (2 x + 3)/2 + C

(iv) Integrate the following with respect to x ,(L x + m) ^(1/2)

**Solution:**

∫ (L x + m) ^(1/2) dx

The given question exactly matches the formula

**∫ (a x + b)^(1/2) dx = (1/a) (a x + b)^(3/2)/(3/2) + C**

Now we are going to integrate the given question by using this formula in the question instead of "a" we have L

So we get,

∫(L x + m) ^(1/2) dx = (1/L) (L x + m)^(3/2)/(3/2) + C

= (2/3L) (L x + m)^(3/2) + C

(iv) Integrate the following with respect to x ,√(4 - 5 x)

**Solution:**

∫ √(4 - 5 x) dx

The given question exactly matches the formula

**∫ (a x + b)^(1/2) dx = (1/a) (a x + b)^(3/2)/(3/2) + C**

Now we are going to integrate the given question by using this formula in the question instead of "a" we have -5

So we get,

∫√(4 - 5 x) dx = (1/-5) (4 - 5 x)^(3/2)/(3/2) + C

= (-2/15) (4 - 5 x)^(3/2) + C

integration worksheet2 solution10 integration worksheet2 solution10

- 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