Evaluate :

OR

Evaluate :

Given:

Let x + a = t

Differentiating with respect to x, we get

We know that,

sin(x – y) = sin x cos y – cos x sin y

∵ cos 2a and sin 2a are treated as a constant terms

= cos 2a[t] – sin 2a [log|sin t|] + C

Replace t by x + a, we get

= cos 2a(x + a) – sin 2a [log|x + a|] + C

OR

Let

So, here we try to make the 2 + 6x so that we can easily integrate [We multiply and divide by 6]

[Add and subtract 2]

∴ I = I₁ – I₂ …(i)

Solving I₁

Let 1 + 2x + 3x² = t

Differentiating both the sides, we get

Thus, our equation becomes

[Substitute the value of dx]

[using 1 + 2x + 3x² = t]

Solving I₂

[Taking 3 common]

[Here we add and subtract (1/3)²]

Now, we know that

Now, putting the values of I₁ and I₂ in eq. (i), we get