Evaluate the following integral:
Let us assume …………………………equation 1
Let x= tan θ thus
Differentiating both sides, we get,
Thus substituting old limits, we get a new upper limit and lower limit
For 1 = tan θ
For 0 = tan θ
0 = θ
substitute the values in equation 1
we get …………………….equation 2
trigonometric identity we know
Thus substituting in equation 2 we have
………………………equation 3
By property, we know that
Thus
.....equation 4
Trigonometric formula:
Thus
We know by trigonometric property:
thus
Substituting in equation 4
We know
Thus
......equation 6
We know
Adding equation 3 and equation 6
2 +
Thus
2
2
2
We know b and a being the upper and lower limits respectively.