Find an anti-derivative (or integral) of the following functions by the method of inspection: 1. sin 2x 2. cos 3x 5. e^{2x}. Find the following integrals in Exercises 6 to 20: 6. ∫... Find an anti-derivative (or integral) of the following functions by the method of inspection: 1. sin 2x 2. cos 3x 5. e^{2x}. Find the following integrals in Exercises 6 to 20: 6. ∫(4e^{3x} + 1) dx 7. ∫(x^{2} - rac{1}{x^{2}}) dx ... Choose the correct answer in Exercises 21 and 22: 21. The anti-derivative of (√x + rac{1}{√x}) equals ?
Understand the Problem
The question is asking to find the anti-derivatives or integrals of various functions using different methods of inspection as outlined in the exercise. It focuses on understanding basic integration techniques in calculus.
Answer
$$ \frac{2}{3} x^{3/2} + 2x^{1/2} + C $$
Answer for screen readers
The anti-derivative of $ \sqrt{x} + \frac{1}{\sqrt{x}} $ is: $$ \frac{2}{3} x^{3/2} + 2x^{1/2} + C $$
Steps to Solve
- Identifying the Function to Integrate
We need to find the anti-derivative (integral) of the function $f(x) = \sqrt{x} + \frac{1}{\sqrt{x}}$.
- Breaking Down the Integral
Rewrite the function: $$ f(x) = x^{1/2} + x^{-1/2} $$
- Applying the Power Rule of Integration
Using the power rule of integration: $$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$ We apply this rule to each term of the function:
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For the first term $x^{1/2}$: $$ \int x^{1/2} , dx = \frac{x^{1/2 + 1}}{1/2 + 1} = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} $$
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For the second term $x^{-1/2}$: $$ \int x^{-1/2} , dx = \frac{x^{-1/2 + 1}}{-1/2 + 1} = \frac{x^{1/2}}{1/2} = 2x^{1/2} $$
- Combining the Results
Now, combine the results of the integrations: $$ \int \left( x^{1/2} + x^{-1/2} \right) , dx = \frac{2}{3} x^{3/2} + 2x^{1/2} + C $$
The anti-derivative of $ \sqrt{x} + \frac{1}{\sqrt{x}} $ is: $$ \frac{2}{3} x^{3/2} + 2x^{1/2} + C $$
More Information
The power rule for integration simplifies finding the integral of polynomial terms. Integrating terms with fractional exponents follows the same principle, providing a straightforward approach to solving for anti-derivatives.
Tips
- Forgetting to add the constant of integration $C$ at the end.
- Misapplying the power rule, especially in identifying the new exponent after integration.
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