Tekhnic-Integration-by-part-1-Find-x-sec-2-x-dx-2-Find-x-e-2x-dx-3-Find-ln-x-dx-4-Find-x-2-e-2x-dx-5-Find-e-x-cos-x-dx-

Question Number 156008 by zainaltanjung last updated on 07/Oct/21

Tekhnic Integration by part

1) Find \int x . \sec^{2} x dx

2) Find \int x . e^{2 x} dx

3) Find \int \ln x dx

4) Find \int x^{2} . e^{2 x} dx

5) Find \int e^{x} \cos x dx

Commented by SANOGO last updated on 07/Oct/21

c o o l

Commented by puissant last updated on 07/Oct/21

1)

A = \int x {s e c}^{2} x d x = x t a n x - \int t a n x

= x t a n x - (- l n ∣ c o s x ∣) + C

\Rightarrow A = x t a n x + l n ∣ c o s x ∣ + C . .

Commented by puissant last updated on 07/Oct/21

2)

S = \int x e^{2 x} d x = \frac{1}{2} x e^{2 x} - \frac{1}{2} \int e^{2 x} d x

= \frac{1}{2} x e^{2 x} - \frac{1}{4} e^{2 x} + C

\Rightarrow S = \frac{1}{2} {x e^{2 x} - \frac{1}{2} e^{2 x}} + C

Commented by SANOGO last updated on 07/Oct/21

3) \int l n x = x l n x - x + c

Commented by puissant last updated on 07/Oct/21

3)

I = \int l n x d x

{\begin{cases} u = l n x \\ v^{'} = 1 \end{cases} \Rightarrow {\begin{cases} u^{'} = \frac{1}{x} \\ v = x \end{cases}

\Rightarrow I = x l n x - \int 1 d x

\Rightarrow I = x l n x - x + C

Commented by puissant last updated on 07/Oct/21

4)

J = \int x^{2} e^{2 x} d x

{\begin{cases} u = x^{2} \\ v^{'} = e^{2 x} \end{cases} \Rightarrow {\begin{cases} u^{'} = 2 x \\ v = \frac{1}{2} e^{2 x} \end{cases}

\Rightarrow J = \frac{x^{2}}{2} e^{2 x} - \int {x e}^{2 x} d x

\Rightarrow J = \frac{x^{2}}{2} e^{2 x} - \frac{x}{2} e^{2 x} + \frac{1}{4} e^{2 x} + C

Commented by puissant last updated on 07/Oct/21

5)

D = \int e^{x} c o s x d x

{\begin{cases} u = c o s x \\ v^{'} = e^{x} \end{cases} \Rightarrow {\begin{cases} u^{'} = - s i n x \\ v = e^{x} \end{cases}

\Rightarrow D = e^{x} c o s x + \int e^{x} s i n x d x

{\begin{cases} u = s i n x \\ v^{'} = e^{x} \end{cases} \Rightarrow {\begin{cases} u^{'} = c o s x \\ v = e^{x} \end{cases}

\Rightarrow D = e^{x} c o s x + e^{x} s i n x - \int e^{x} c o s x d x

\Rightarrow D = e^{x} (c o s x + s i n x) - D

\Rightarrow 2 D = e^{x} (c o s x + s i n x)

\Rightarrow D = \frac{e^{x}}{2} (c o s x + s i n x) + C . .

Commented by tabata last updated on 08/Oct/21

3) w i t h o u t b y p a r t

\int l n x d x = \int (l n x + 1 - 1) d x

= \int ((l n x + 1) - 1) d x

= \int d (x l n x) - \int d x

= x l n x - x + C

M

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