Q 63. Solve the integral:聽鈭玡-x聽cos... [FREE SOLUTION]

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Chapter 5: Q 63. (page 429)

Solve the integral: $鈭� e^{- x} \cos x d x$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

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

Step 2. Solve the integration by parts.

We have,

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

and

$d v = \frac{d x}{e^{x}} d v = \frac{1}{e^{x}} d x v = 鈭� \frac{1}{e^{x}} d x v = - \frac{1}{e^{x}}$

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

role="math" localid="1648892352076" $鈭� e^{- x} \cos x d x = (\cos x (- \frac{1}{e^{x}}) - 鈭� (- \frac{1}{e^{x}}) (- \sin x) d x) = (- 鈭� \frac{1}{e^{x}} \sin x d x - \frac{1}{e^{x}} \cos x)$

Step 3. Integration by parts.

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

and

$d v = \frac{d x}{e^{x}} d v = \frac{1}{e^{x}} d x v = 鈭� \frac{1}{e^{x}} d x v = - \frac{1}{e^{x}}$

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

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91影视

Solve the integral: $鈭� e^{- x} \cos x d x$

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�e-xcosxdx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integration by parts.

Step 3. Integration by parts.

One App. One Place for Learning.

Most popular questions from this chapter

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Solve the integral: $鈭� e^{- x} \cos x d x$