**I. Evaluate the following:**

**Question 1.∫ log x dxSolution:**

**Question 2.∫ sin 3x dxSolution:**

**Question 3.∫x x dxSolution:**

**Question 4.∫ x dxSolution:**

**Question 5.**

**Question 6.**

**Question 7.**

**Question 8.∫x . x dxSolution:**

**Question 9.∫ log x dxSolution:**

**Question 10.∫ cos 3x dxSolution:**

**Question 11.∫x x dxSolution:**

**Question 12.∫ x dxSolution:**

**Question 13.Solution:**

= t(log t – 1) + c

= (log x) . [log(log x) – 1] + c.

**Question 14.Solution:**

**Question 15.**

**Question 16.∫sin θ . log(cos θ) dθSolution:**

Let I = ∫sin θ . log (cos θ) dθ

= ∫log(cos θ) . sin θ dθ

Put cos θ = t

∴ -sin θ dθ = dt

∴ sin θ dθ = -dt

= -t log t + t + c

= -cos θ . log(cos θ) + cos θ + c

= -cos θ [log(cos θ) – 1] + c.

**Question 17.**

**Question 18.Solution:**

**Question 19.**

**Question 20.**

**Question 21.**

**II. Integrate the following functions w.r.t. x:**

**Question 1. sin 3xSolution:**

**Question 2.cos 2xSolution:**

**Question 3.sin(log x)Solution:**

**Question 4.**

**Question 5.Solution:**

**Question 6.Solution:**

**Question 7.Solution:**

**Question 8.**

**Question 9.**

= A[-4 – 2x] + B

= -2Ax + (B – 4A)

Comparing the coefficients of x and the constant term on both sides, we get

-2A = 1, B – 4A = 0

**Question 10.Solution:**

**Question 11.Solution:**

**Question 12.Solution:**

**III. Integrate the following functions w.r.t. x:**

**Question 1.**

**Question 2.Solution:**

**Question 3.**

**Question 4.Solution:**

**Question 5.Solution:**

**Question 6.**

**Question 7.Solution:**

**Question 8.**

**Question 9.**