Q. 26 Use whatever method you like to ... [FREE SOLUTION]

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Chapter 5: Q. 26 (page 496)

Use whatever method you like to solve each of the given definite and indefinite integrals. These integrals are neither in order of difficulty nor in order of technique. Many of the integrals can be solved in more than one way.
$鈭� {(\sin x + e^{x})}^{2} d x$

Short Answer

Expert verified

The result is $\frac{1}{2} (x - \frac{1}{2} \sin (2 x)) + e^{x} \sin (x) - e^{x} \cos (x) + \frac{1}{2} e^{2 x} + C$ .

Step by step solution

Step 1. Given information.

Consider the given integral.

$鈭� {(\sin x + e^{x})}^{2} d x$

Step 2. Find the integral.

Simplify the given integral.

$\begin{array}{rcl} 鈭� {(\sin x + e^{x})}^{2} d x & = & 鈭� (\sin^{2} x + 2 e^{x} \sin x + e^{2 x}) d x \\ = & 鈭� \sin^{2} x d x + 鈭� 2 e^{x} \sin x d x + 鈭� e^{2 x} d x \\ = & \frac{1}{2} 鈭� (1 - \cos 2 x) d x + 2 鈭� e^{x} \sin x d x + [\frac{e^{2 x}}{2}] \\ = & \frac{1}{2} (x - \frac{\sin 2 x}{2}) d x + [\frac{e^{2 x}}{2}] + 2 鈭� e^{x} \sin x d x \end{array}$

Step 3. Solve the integral 2∫exsinxdx.

Use the by part's method to find the integral.

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

Step 4. Conclusion.

Substitute the value of integral and simplify.

$\begin{array}{rcl} 鈭� {(\sin x + e^{x})}^{2} d x & = & \frac{1}{2} (x - \frac{\sin 2 x}{2}) d x + [\frac{e^{2 x}}{2}] + 2 (\frac{- e^{x} \cos x + e^{x} \sin x}{2}) + C \\ = & \frac{1}{2} (x - \frac{\sin 2 x}{2}) d x + [\frac{e^{2 x}}{2}] - e^{x} \cos x + e^{x} \sin x + C \end{array}$

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91影视

Use whatever method you like to solve each of the given definite and indefinite integrals. These integrals are neither in order of difficulty nor in order of technique. Many of the integrals can be solved in more than one way.
$鈭� {(\sin x + e^{x})}^{2} d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the integral.

Step 3. Solve the integral 2∫exsinxdx.

Step 4. Conclusion.

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Use whatever method you like to solve each of the given definite and indefinite integrals. These integrals are neither in order of difficulty nor in order of technique. Many of the integrals can be solved in more than one way. 鈭�sinx+ex2dx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the integral.

Step 3. Solve the integral 2∫exsinxdx.

Step 4. Conclusion.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Geometry

Decision Maths

Pure Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use whatever method you like to solve each of the given definite and indefinite integrals. These integrals are neither in order of difficulty nor in order of technique. Many of the integrals can be solved in more than one way.
$鈭� {(\sin x + e^{x})}^{2} d x$