In this page integration worksheet2 solution9 we are going to see
solution of some practice question from the worksheet of integration.

**Question 9**

(i) Integrate the following with respect to x , 1/cos² (px + a)

**Solution:**

∫ 1/cos² (px + a) dx

1/cos² x can be written as sec² x

∫ 1/cos² (px + a) dx = ∫sec² (px + a) dx

**∫ sec****² (a x + b) dx = (1/a) tan (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 p

So we get,

∫sec²(p x + a) dx = (1/p) tan (p x + a) + C

= tan (p x + a)/p + C

(ii) Integrate the following with respect to x , 1/sin² (L - m x)

**Solution:**

∫ 1/sin² (L - m x) dx

1/sin² (L - m x) can be written as cosec² x

∫ 1/sin² (L - m x) dx = ∫cosec² (L - m x) dx

**∫ cosec****² (a x + b) dx = -(1/a) cot (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 -m

So we get,

∫cosec² (L - m x) dx = - (1/-m) cot (L - m x) + C

= cot (L - m x)/m + C

(iii) Integrate the following with respect to x , (a x +b)⁻⁸

**Solution:**

∫ (a x +b)⁻⁸ dx

**∫ x^n**** dx = x^(n + 1)/(n + 1) + C**

Now we are going to integrate the given question by using this formula in the question instead of "n" we have -8 and instead of "a" we have "a" itself

So we get,

∫(a x +b)⁻⁸ dx = (1/a) (a x + b)^(-8 + 1)/(-8 + 1) + C

= (1/a) (a x + b)^(-7)/(-7) + C

= -(a x + b)^-7/(7a) + C

(iv) Integrate the following with respect to x , (3 - 2 x)⁻¹

**Solution:**

∫ (3 - 2 x)⁻¹ dx = ∫ 1/(3 - 2 x) dx

**∫ 1/(a x + b)**** dx = (1/a) log (a x + b) + C**

Now
we are going to integrate the given question by using this formula in
the question instead of "n" we have -1 and instead of "a" we have "-2".

So we get,

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

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

(v) Integrate the following with respect to x , e^-x

**Solution:**

∫ e^-x dx = e^-x/(-1) + C

= - e^-x + C

integration worksheet2 solution9 integration worksheet2 solution9

- 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