Evaluate:

OR

Evaluate:

Let

We know that,

cos 2x = 2cos²x – 1

and sin 2x = 2sinxcosx

∴ We can write,

…(i)

⇒ I = I₁ + I₂

Solving I₁

Using Integration by parts, taking e^2x as first function and sec²x as second function

We know that,

Putting the value of I₁ in eq. (i)

OR

Let

We can write

Dividing numerator by denominator as follows:

Hence,

Dividend = Divisor × Quotient + Remainder

x⁴ = (x³ – x² + x – 1)(x + 1) + 1

Thus,

…(A)

We can write this as

⇒ 1 = Ax(x – 1) + B(x – 1) + C(x² + 1)

⇒ 1 = Ax² – Ax + Bx – B + Cx² + C

⇒ 1 = x² (A + C) + x (- A + B) + 1 (- B + C)

On comparing the coefficients of x and constant terms from both sides, we get

A + C = 0 …(i)

-A + B = 0 …(ii)

-B + C = 1 …(iii)

From eq. (i) and (ii), we get

A + C – A + B = 0

⇒ B + C = 0 …(iv)

Now, from eq. (iii) and (iv), we get

- B + C + B + C = 1 + 0

⇒ 2C = 1

Putting the value of C in eq. (iv), we get

Now, putting the value of C in eq. (i), we get

Hence, we can write

Therefore, eq(A) become

Integrating with respect to x, we get

Solving I₁

…(i)

Put t = x² + 1

Differentiate with respect to x

Putting the value of xdx in eq. (i)

Solving I₂

Solving I₃

Hence,