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प्रश्न
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
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उत्तर
Let I = `int log (1 + x)^((1 + x)).dx`
= `int (1 + x)log(1 + x).dx`
= `int [log(1 + x)] (1 + x).dx`
= `[log(1 + x) int (1 + x).dx - int[d/dt {log(1 + x)} int (1 + x).dx].dx`
= `[log (1 + x)] [(1 + x)^2/2] - int 1/(x + 1).(x + 1)^2/(2).dx`
= `(x + 1)^2/(2).log(1 + x) - (1)/(2) int (x + 1).dx`
= `(x + 1)^2/(2).log (1 + x) - (1)/(2).(x + 1)^2/(2) + c`
= `(x + 1)^2/(2)[log (1 + x) - 1/2] + c`.
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