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प्रश्न
Evaluate the following integrals using properties of integration:
`int_0^1 (log(1 + x))/(1 + x^2) "d"x`
उत्तर
Let I = `int_0^1 (log(1 + x))/(1 + x^2) "d"x`
Now, on putting x = tanθ
dx = sec2θ dθ
| x | 0 | 1 |
| θ | 0 | `pi/4` |
I = `int_0^(pi/4) (log(1 + tan theta))/(1 + tan^2theta) sec^2 theta "d"theta`
I = `int_0^(pi/4) (log(1 + tan theta))/(sec^2theta) sec^2theta "d"theta`
`int_0^"a" f(x) "d"x = int_0^"a" f("a" - x) "d"x`
I = `int_0^(pi/4) log(1 + tan(pi/4 - theta)) "d"theta`
= `int_0^(pi/4) log[1 + (tan pi/4 - tan theta)/(1 + tan pi/4 tan theta)] "d"theta`
= `int_0^(pi/4) log(1 + (1 - tan theta)/(1 + tan theta))"d"theta`
= `int_0^(pi/4) log ((1 + tan theta + 1 - tan theta)/(1 + tan theta)) "d"theta`
= `int_0^(pi/4) log (2/(1 + tan theta)) "d"theta`
= `int_0^(pi/4) [log2 - log(1 + tan theta)] "d"theta`
= `log 2 int_0^(pi/4) "d"theta - int_0^(pi/4) log(1 + tan theta) "d"theta`
I = `log 2(pi/4) - "I"`
2I = `pi/4 log 2`
I = `pi/8 log 2`
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