Evaluate the following integrals –
Let
Let us assume
We know and derivative of a constant is 0.
⇒ 2x + 5 = λ(0 – 4 – 3×2x2-1) + μ
⇒ 2x + 5 = λ(–4 – 6x) + μ
⇒ 2x + 5 = –6λx + μ – 4λ
Comparing the coefficient of x on both sides, we get
–6λ = 2
Comparing the constant on both sides, we get
μ – 4λ = 5
Hence, we have
Substituting this value in I, we can write the integral as
Let
Now, put 10 – 4x – 3x2 = t
⇒ (–4 – 6x)dx = dt (Differentiating both sides)
Substituting this value in I1, we can write
Recall
Let
We can write 10 – 4x – 3x2 = –(3x2 + 4x – 10)
Hence, we can write I2 as
Recall
Substituting I1 and I2 in I, we get
Thus,