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Magazine Chapter 5: Problem 25
Evaluate the integrals using appropriate substitutions. $$ \int t \sqrt{7 t^{2}+12} d t $$
Short Answer
Expert verified
The integral is \( \frac{1}{21} (7t^{2} + 12)^{3/2} + C \).
Step by step solution
01
Identify the Integral Type
The integral \( \int t \sqrt{7 t^{2}+12} \, dt \) is a function of \( t \) multiplied by the square root of another function. This suggests that a substitution based on the inner function of the square root might be useful.
02
Choose a Suitable Substitution
To simplify the integral, we can use the substitution \( u = 7t^{2} + 12 \). This choice will simplify the square root and the expression under the derivative. Note that the derivative of \( u \) with respect to \( t \) is needed next.
03
Derive the Expression for du
Differentiate \( u = 7t^{2} + 12 \) with respect to \( t \): \( \frac{du}{dt} = 14t \). This implies \( du = 14t \, dt \). Solve for \( t \, dt \): \( t \, dt = \frac{1}{14} du \).
04
Substitute into the Integral
Replace \( t \, dt \) in the original integral with \( \frac{1}{14} du \), and replace \( \sqrt{7t^2 + 12} \) with \( \sqrt{u} \). The integral becomes: \( \int \sqrt{u} \cdot \frac{1}{14} \, du = \frac{1}{14} \int u^{1/2} \, du \).
05
Integrate with Respect to u
Use the power rule for integration: \( \int u^{n} \, du = \frac{u^{n+1}}{n+1} + C \). Applying this to \( u^{1/2} \), we get \( \int u^{1/2} \, du = \frac{u^{3/2}}{3/2} + C = \frac{2}{3} u^{3/2} + C \). Substituting back, we have \( \frac{1}{14} \cdot \frac{2}{3} u^{3/2} + C = \frac{1}{21} u^{3/2} + C \).
06
Substitute Back to the Original Variable
Recall the substitution \( u = 7t^{2} + 12 \). Replace \( u \) in the integrated expression: \( \frac{1}{21} (7t^{2} + 12)^{3/2} + C \).
07
Write the Final Answer
The solution to the integral is \( \frac{1}{21} (7t^{2} + 12)^{3/2} + C \). This is the evaluated integral rewritten in terms of the original variable \( t \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integral Calculus
Integral calculus is a fundamental part of calculus that focuses on finding the antiderivative or the integral of functions.
It provides a way to calculate areas under curves, among other applications.
In the example provided, substitution helps simplify the process by allowing the integrand to be expressed in terms of a new variable, which can make the integral more straightforward to evaluate.
It provides a way to calculate areas under curves, among other applications.
- An integral can be definite (with limits) or indefinite (without limits).
- Indefinite integrals include an arbitrary constant, often represented as "C," since integration is the reverse process of differentiation and any constant vanishes when differentiated.
- A basic goal of integral calculus is to find a function whose derivative matches the original function in the integrand.
In the example provided, substitution helps simplify the process by allowing the integrand to be expressed in terms of a new variable, which can make the integral more straightforward to evaluate.
Substitution Method
The substitution method is a valuable technique used to simplify integrals, especially when dealing with composite functions.
It's akin to the chain rule for differentiation but in reverse.
This step is crucial since it turns a complex product and radical expression into a form that can be more easily tackled using the power rule for integration.
It's akin to the chain rule for differentiation but in reverse.
- To use substitution, you identify a part of the integral that can be replaced with a single variable, often aiming to simplify the calculation.
- For the given integral \( \int t \sqrt{7t^2 + 12} \, dt \), the substitution \( u = 7t^2 + 12 \) simplifies the expression.
- By differentiating \( u \), we get \( \frac{du}{dt} = 14t \), which can be rearranged to express the measure \( dt \).
This step is crucial since it turns a complex product and radical expression into a form that can be more easily tackled using the power rule for integration.
Power Rule for Integration
The power rule for integration is a straightforward method used to find the antiderivative of a function with respect to a variable.
It is particularly simple: if you have \( u^n \), where \( n eq -1 \), its integral is \( \frac{u^{n+1}}{n+1} + C \).
In any calculus problem, understanding and using these rules can greatly aid in evaluating integrals more efficiently.
It is particularly simple: if you have \( u^n \), where \( n eq -1 \), its integral is \( \frac{u^{n+1}}{n+1} + C \).
- This rule applies when the function is expressed as a power of a single variable, which is why using substitution to isolate such expressions is helpful.
- In the stepped solution, after substitution, the integral becomes \( \int u^{1/2} \, du \), allowing the power rule to be easily applied.
- Thus, integrating \( u^{1/2} \) gives \( \frac{u^{3/2}}{3/2} + C \), which simplifies further by multiplication to \( \frac{2}{3} u^{3/2} + C \).
In any calculus problem, understanding and using these rules can greatly aid in evaluating integrals more efficiently.
