Q 75. Solve the definite integral.鈭�0... [FREE SOLUTION]

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Chapter 5: Q 75. (page 429)

Solve the definite integral.
${鈭�}_{0}^{蟺} e^{x} \sin x d x$

Short Answer

Expert verified

The solution is $\frac{1}{2} (e^{蟺} + 1)$ .

Step by step solution

Step 1. Given information.

The given integral is ${鈭�}_{0}^{蟺} e^{x} \sin x d x$ .

Step 2. First, solve the indefinite integral.

$\begin{array}{rcl} I & = & 鈭� e^{x} \sin x d x \\ = & \sin x 鈭� e^{x} d x - 鈭� [\frac{d}{d x} (\sin x) 鈭� e^{x} d x] d x \\ = & \sin x e^{x} - 鈭� \cos x e^{x} d x \\ = & \sin x e^{x} - [\cos x 鈭� e^{x} d x - 鈭� \{\frac{d}{d x} (\cos x) 鈭� e^{x} d x\} d x] \\ = & \sin x e^{x} - [\cos x e^{x} - 鈭� - \sin x e^{x} d x] \\ = & \sin x e^{x} - \cos x e^{x} - 鈭� \sin x e^{x} d x \\ = & e^{x} (\sin x - \cos x) - 鈭� \sin x e^{x} d x \\ = & e^{x} (\sin x - \cos x) - I \\ I & = & e^{x} (\sin x - \cos x) - I \end{array}$

Step 3. Continued the above solution.

$\begin{array}{rcl} I & = & e^{x} (\sin x - \cos x) - I \\ I + I & = & e^{x} (\sin x - \cos x) \\ 2 I & = & e^{x} (\sin x - \cos x) \\ I & = & \frac{1}{2} e^{x} (\sin x - \cos x) \end{array}$

Step 4. Now apply the limit and solve the integral.

$\begin{array}{rcl} {鈭�}_{0}^{蟺} e^{x} \sin x d x & = & {[\frac{1}{2} e^{x} (\sin x - \cos x)]}_{0}^{蟺} \\ = & [\frac{1}{2} e^{蟺} (蝉颈苍蟺 - 肠辞蝉蟺) - \frac{1}{2} e^{0} (\sin 0 - \cos 0)] \\ = & \frac{1}{2} e^{蟺} (0 - (- 1)) - \frac{1}{2} (0 - 1) 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� \{肠辞蝉蟺 = - 1, \cos 0 = 1\} \\ = & \frac{1}{2} e^{蟺} + \frac{1}{2} \\ = & \frac{1}{2} (e^{蟺} + 1) \end{array}$

Step 5. Simplified answer.

The value of the given integral is $\frac{1}{2} (e^{蟺} + 1)$ .

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91影视

Solve the definite integral.
${鈭�}_{0}^{蟺} e^{x} \sin x d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. First, solve the indefinite integral.

Step 3. Continued the above solution.

Step 4. Now apply the limit and solve the integral.

Step 5. Simplified answer.

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Solve the definite integral.鈭�0蟺exsinxdx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. First, solve the indefinite integral.

Step 3. Continued the above solution.

Step 4. Now apply the limit and solve the integral.

Step 5. Simplified answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Probability and Statistics

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Theoretical and Mathematical Physics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Solve the definite integral.
${鈭�}_{0}^{蟺} e^{x} \sin x d x$