Evaluate the following integral:

  [CBSE 2011]

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.