Q. 45.0( 1 Vote )

# Evaluate the following integrals –

Answer :

Let

Let us assume

We know and derivative of a constant is 0.

⇒ x + 2 = λ(2x^{2-1} + 1 + 0) + μ

⇒ x + 2 = λ(2x + 1) + μ

⇒ x + 2 = 2λx + λ + μ

Comparing the coefficient of x on both sides, we get

2λ = 1 ⇒

Comparing the constant on both sides, we get

λ + μ = 2

Hence, we have

Substituting this value in I, we can write the integral as

Let

Now, put x^{2} + x + 1 = t

⇒ (2x + 1)dx = dt (Differentiating both sides)

Substituting this value in I_{1}, we can write

Recall

Let

We can write

Hence, we can write I_{2} as

Recall

Substituting I_{1} and I_{2} in I, we get

Thus,

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