Q 59. Solve the integral:聽鈭玡2x聽sin... [FREE SOLUTION]

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Chapter 5: Q 59. (page 429)

Solve the integral: $鈭� e^{2 x} \sin x d x$

Short Answer

Expert verified

The required answer is $- \frac{1}{5} e^{2 x} \cos (x) + \frac{2}{5} e^{2 x} \sin x + c$ .

Step by step solution

Step 1. Given information.

We have given integral is $鈭� e^{2 x} \sin x d x$ .

Step 2. Solve the integration by parts .

We have,

$u = \sin x d u = \cos x d x$

and

$d v = e^{2 x} d x v = 鈭� e^{2 x} d x v = \frac{e^{2 x}}{2}$

The formula of integration by parts is $鈭� u d v = u v - 鈭� v d u$ .

role="math" localid="1648913674528" $鈭� e^{2 x} \sin x d x = (\sin (x) \frac{e^{2 x}}{2} - 鈭� \frac{e^{2 x}}{2} \cos (x) d x) = (\frac{e^{2 x}}{2} \sin (x) - 鈭� \frac{e^{2 x}}{2} \cos (x) d x) = \frac{e^{2 x}}{2} \sin (x) - \frac{1}{2} 鈭� e^{2 x} \cos (x) d x$

Step 3. Integration by parts.

We have,

$u = \cos x d u = - \sin x d x$

and

$d v = e^{2 x} d x v = 鈭� e^{2 x} d x v = \frac{e^{2 x}}{2}$

So,

$\frac{e^{2 x}}{2} \sin (x) - \frac{1}{2} 鈭� e^{2 x} \cos (x) d x = \frac{e^{2 x}}{2} \sin (x) - \frac{1}{2} (\cos x \frac{e^{2 x}}{2} - 鈭� \frac{e^{2 x}}{2} (- \sin (x)) d x) = \frac{e^{2 x}}{2} \sin (x) - \frac{1}{2} (\frac{e^{2 x}}{2} \cos x + 鈭� \frac{e^{2 x}}{2} \sin (x) d x) = \frac{e^{2 x}}{2} \sin (x) - \frac{e^{2 x}}{4} \cos x - \frac{1}{4} 鈭� e^{2 x} \sin (x) d x = - \frac{e^{2 x}}{5} \cos (x) + \frac{2}{5} e^{2 x} \sin x + c$

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91影视

Solve the integral: $鈭� e^{2 x} \sin x d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integration by parts .

Step 3. Integration by parts.

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91影视

Solve the integral:鈭�e2xsinxdx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integration by parts .

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Most popular questions from this chapter

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Solve the integral: $鈭� e^{2 x} \sin x d x$