sin-ln-x-dx-

Question Number 156695 by amin96 last updated on 14/Oct/21

\int s i n (l n (x)) d x = ?

Answered by puissant last updated on 10/Nov/21

Q = \int_{0}^{1} s i n (l n x) d x; u = l n x \to x = e^{u} \to d x = e^{u} d u

\Rightarrow Q = \int_{0}^{1} e^{u} s i n u d u

I B P : {\begin{cases} i = s i n u \\ j^{'} = e^{u} \end{cases} \Rightarrow {\begin{cases} i^{'} = c o s u \\ j = e^{u} \end{cases}

\Rightarrow Q = {[e^{u} s i n u]}_{0}^{1} - \int_{0}^{1} e^{u} c o s u d u

D o u b l e I B P : {\begin{cases} i = c o s u \\ j^{'} = e^{u} \end{cases} \Rightarrow {\begin{cases} i^{'} = - s i n u \\ j = e^{u} \end{cases}

\Rightarrow Q = {[e^{u} s i n u - e^{u} c o s u]}_{0}^{1} - \int_{0}^{1} e^{u} s i n u d u

\Rightarrow Q = {\frac{e^{u}}{2} (s i n u - c o s u)}_{0}^{1}

\Rightarrow Q = {\frac{e}{2} (s i n (1) - c o s (1))} + \frac{1}{2}

∴∵ Q = \frac{1}{2} {e (s i n (1) - c o s (1)) + 1} . .

\dots \dots \dots \dots \dots \dots L e p u i s s a n t \dots \dots \dots \dots \dots . .

Commented by amin96 last updated on 14/Oct/21

v e r y g o o d . t h a n k s s i r

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