Problem 11
Question
Find the indefinite integral and check the result by differentiation. $$ \int \sqrt{4 x^{2}-5}(8 x) d x $$
Step-by-Step Solution
Verified Answer
The indefinite integral of \( \sqrt{4x^{2}-5} \times 8x \) is \( \frac{2}{3}(4x^{2}-5)^{3/2}+C \). The result has been verified with differentiation.
1Step 1: Substitution
Let a substitution \( u=4x^{2}-5 \), then the derivative of \( u \) is \( du=8xdx \). The given integral can be rewritten according to this substitution, and we have \(\int u^{1/2} du \).
2Step 2: Evaluate the Integral
We can now evaluate the integral \( \int u^{1/2} du \) = \(\frac{2}{3}u^{3/2}+C \), where \( C \) is the constant of integration.
3Step 3: Back Substitution
Now, use back substitution to replace \( u \) with \( 4x^{2}-5 \), therefore we get \( \frac{2}{3}(4x^{2}-5)^{3/2}+C \).
4Step 4: Check the Result by Differentiation
We can check our result by differentiating \( \frac{2}{3}(4x^{2}-5)^{3/2} \). Using the chain rule, we obtain \( 8x\sqrt{4x^{2}-5} \), which is the original integrand. This confirms that our solution is correct.
Key Concepts
Substitution MethodDifferentiationChain Rule
Substitution Method
The substitution method is a useful technique in integration, aimed at simplifying an integrand by changing the variable of integration. This method is particularly handy when dealing with complex expressions or composite functions. In the given exercise, we started by setting a new variable \( u = 4x^2 - 5 \). The purpose of this substitution is to make the integral easier to solve.
When we substitute, it's essential to also replace \( dx \) with \( du \). This involves finding the derivative of \( u \) with respect to \( x \), which gives us \( du = 8x \, dx \). This substitution transforms the original integral into a format that is more straightforward: \( \int u^{1/2} \, du \).
When we substitute, it's essential to also replace \( dx \) with \( du \). This involves finding the derivative of \( u \) with respect to \( x \), which gives us \( du = 8x \, dx \). This substitution transforms the original integral into a format that is more straightforward: \( \int u^{1/2} \, du \).
- Choose a substitution \( u \) that simplifies the integrand.
- Find \( du \) by differentiating \( u \) with respect to \( x \).
- Rewrite the integral in terms of \( u \) and \( du \).
Differentiation
Differentiation is the process of finding the derivative of a function, which represents the rate at which the function is changing. This concept is vital for verifying the correctness of an integral solution by differentiation.
In the final step of the exercise, we check our result by differentiating the obtained function \( \frac{2}{3}(4x^{2}-5)^{3/2} \). Differentiating ensures that when we differentiate our indefinite integral, we get back to the original integrand. This is where the rules of differentiation, such as the power rule and the chain rule, come into play. A successful differentiation will yield the integrand, confirming that our integration process was accurate. Differentiation steps include:
In the final step of the exercise, we check our result by differentiating the obtained function \( \frac{2}{3}(4x^{2}-5)^{3/2} \). Differentiating ensures that when we differentiate our indefinite integral, we get back to the original integrand. This is where the rules of differentiation, such as the power rule and the chain rule, come into play. A successful differentiation will yield the integrand, confirming that our integration process was accurate. Differentiation steps include:
- Apply the power rule to derive the function.
- Incorporate the chain rule where necessary to correctly obtain the derivative.
- Compare the differentiated result with the original integrand.
Chain Rule
The chain rule is a foundational concept in calculus used to differentiate composite functions. It is particularly relevant when confirming the result of an integration problem like this one, where composite functions are involved.
During the differentiation step, the chain rule assists by allowing us to handle nested functions in derivatives. For \( \frac{2}{3}(4x^{2}-5)^{3/2} \), a composite function, the chain rule guides us to differentiate the outer function and then multiply by the derivative of the inner function \( 4x^2 - 5 \). The application of the chain rule results in a complete derivative: \( 8x\sqrt{4x^{2}-5} \), matching our original integrand of \( 8x\sqrt{4x^{2}-5} \).
During the differentiation step, the chain rule assists by allowing us to handle nested functions in derivatives. For \( \frac{2}{3}(4x^{2}-5)^{3/2} \), a composite function, the chain rule guides us to differentiate the outer function and then multiply by the derivative of the inner function \( 4x^2 - 5 \). The application of the chain rule results in a complete derivative: \( 8x\sqrt{4x^{2}-5} \), matching our original integrand of \( 8x\sqrt{4x^{2}-5} \).
- Identify the outer and inner functions within a composite function.
- Differentiate the outer function while keeping the inner function unchanged.
- Multiply by the derivative of the inner function for the final result.
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