9. If $f$ Is The Antiderivative Of $\frac{x^2}{1+x^4}$ Such That $f(1)=6$, Find $f(5$\].

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Introduction

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In this problem, we are given a function $f$ that is the antiderivative of $\frac{x^2}{1+x^4}$. We are also given that $f(1)=6$, and we need to find the value of $f(5)$. To solve this problem, we will use the concept of antiderivatives and the fundamental theorem of calculus.

The Fundamental Theorem of Calculus

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The fundamental theorem of calculus states that if $f$ is an antiderivative of $F$, then $f$ is the antiderivative of $F$ if and only if $f(x) = F(x) + C$, where $C$ is a constant. In other words, if we know the antiderivative of a function, we can find the value of the function at any point by plugging in the value of the point and the constant $C$.

Finding the Antiderivative

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To find the antiderivative of $\frac{x^2}{1+x^4}$, we can use the following steps:

1. Substitution Method: We can substitute $u = x^2$ and $du = 2x \, dx$. This will simplify the integral and make it easier to solve.
2. Integration by Parts: We can use integration by parts to integrate the resulting expression.
3. Simplification: We can simplify the resulting expression to find the antiderivative.

Substitution Method

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Let $u = x^2$. Then $du = 2x \, dx$. We can substitute these expressions into the original integral:

$$\int \frac{x^2}{1+x^4} \, dx = \int \frac{u}{1+u^2} \cdot \frac{1}{2u} \, du$$

Integration by Parts

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We can use integration by parts to integrate the resulting expression:

$$\int \frac{u}{1+u^2} \cdot \frac{1}{2u} \, du = \frac{1}{2} \int \frac{1}{1+u^2} \, du$$

Simplification

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We can simplify the resulting expression to find the antiderivative:

$$\frac{1}{2} \int \frac{1}{1+u^2} \, du = \frac{1}{2} \arctan u + C$$

Back Substitution

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We can substitute back $u = x^2$ to find the antiderivative in terms of $x$:

$$\frac{1}{2} \arctan u + C = \frac{1}{2} \arctan x^2 + C$$

Evaluating the Antiderivative

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We are given that $f(1)=6$, and we need to find the value of $f(5)$. We can use the fundamental theorem of calculus to evaluate the antiderivative:

$$f(5) = \frac{1}{2} \arctan 5^2 + C$$

Finding the Constant

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We can use the given information $f(1)=6$ to find the constant $C$:

$$6 = \frac{1}{2} \arctan 1^2 + C$$

Simplification

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We can simplify the resulting expression to find the constant $C$:

$$6 = \frac{1}{2} \arctan 1 + C$$

Evaluating the Constant

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We can evaluate the constant $C$:

$$C = 6 - \frac{1}{2} \arctan 1$$

Finding the Value of $f(5)$

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We can substitute the value of the constant $C$ into the expression for $f(5)$:

$$f(5) = \frac{1}{2} \arctan 5^2 + 6 - \frac{1}{2} \arctan 1$$

Simplification

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We can simplify the resulting expression to find the value of $f(5)$:

$$f(5) = \frac{1}{2} \arctan 25 + 6 - \frac{1}{2} \arctan 1$$

Evaluating the Value of $f(5)$

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We can evaluate the value of $f(5)$:

$$f(5) = \frac{1}{2} \arctan 25 + 6 - \frac{1}{2} \cdot \frac{\pi}{4}$$

Simplification

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We can simplify the resulting expression to find the value of $f(5)$:

$$f(5) = \frac{1}{2} \arctan 25 + 6 - \frac{\pi}{8}$$

Evaluating the Value of $f(5)$

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We can evaluate the value of $f(5)$:

$$f(5) = \frac{1}{2} \cdot 1.5708 + 6 - 0.3927$$

Simplification

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We can simplify the resulting expression to find the value of $f(5)$:

$$f(5) = 0.7854 + 6 - 0.3927$$

Evaluating the Value of $f(5)$

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We can evaluate the value of $f(5)$:

$$f(5) = 5.3927$$

The final answer is $\boxed{5.3927}$.

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Introduction

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In the previous section, we discussed how to find the value of an antiderivative using the fundamental theorem of calculus. In this section, we will answer some common questions related to finding the value of an antiderivative.

Q: What is an antiderivative?

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A: An antiderivative is a function that is the inverse of a derivative. In other words, if we have a function $f(x)$, then its antiderivative is a function $F(x)$ such that $F'(x) = f(x)$.

Q: How do I find the antiderivative of a function?

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A: To find the antiderivative of a function, we can use various techniques such as substitution, integration by parts, and integration by partial fractions. We can also use the fundamental theorem of calculus to find the antiderivative.

Q: What is the fundamental theorem of calculus?

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A: The fundamental theorem of calculus states that if $f$ is an antiderivative of $F$, then $f$ is the antiderivative of $F$ if and only if $f(x) = F(x) + C$, where $C$ is a constant.

Q: How do I use the fundamental theorem of calculus to find the value of an antiderivative?

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A: To use the fundamental theorem of calculus to find the value of an antiderivative, we need to know the antiderivative of the function and the value of the constant $C$. We can then plug in the value of the point and the constant $C$ into the expression for the antiderivative to find the value of the antiderivative at that point.

Q: What is the difference between an antiderivative and an integral?

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A: An antiderivative is a function that is the inverse of a derivative, while an integral is a number that represents the area under a curve. The antiderivative of a function is a function that is the inverse of the derivative of the function, while the integral of a function is a number that represents the area under the curve of the function.

Q: How do I use the antiderivative to find the value of an integral?

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A: To use the antiderivative to find the value of an integral, we need to know the antiderivative of the function and the value of the constant $C$. We can then plug in the value of the point and the constant $C$ into the expression for the antiderivative to find the value of the integral at that point.

Q: What are some common techniques for finding the antiderivative of a function?

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A: Some common techniques for finding the antiderivative of a function include substitution, integration by parts, and integration by partial fractions. We can also use the fundamental theorem of calculus to find the antiderivative.

Q: How do I know if I have found the correct antiderivative?

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A: To know if you have found the correct antiderivative, you need to check if the derivative of the antiderivative is equal to the original function. If it is, then you have found the correct antiderivative.

Q: What are some common mistakes to avoid when finding the antiderivative of a function?

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A: Some common mistakes to avoid when finding the antiderivative of a function include:

• Not checking if the derivative of the antiderivative is equal to the original function
• Not using the correct technique for finding the antiderivative
• Not checking if the antiderivative is a function of the correct variable
• Not checking if the antiderivative is a function of the correct domain

Conclusion

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In this article, we have discussed how to find the value of an antiderivative using the fundamental theorem of calculus. We have also answered some common questions related to finding the value of an antiderivative. By following the techniques and tips outlined in this article, you should be able to find the value of an antiderivative with ease.

Additional Resources

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If you are having trouble finding the value of an antiderivative, you may want to try the following resources:

• Online calculators: There are many online calculators available that can help you find the value of an antiderivative.
• Math textbooks: There are many math textbooks available that can provide you with step-by-step instructions on how to find the value of an antiderivative.
• Online tutorials: There are many online tutorials available that can provide you with video instructions on how to find the value of an antiderivative.

Final Thoughts

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Finding the value of an antiderivative can be a challenging task, but with the right techniques and resources, it can be done with ease. By following the techniques and tips outlined in this article, you should be able to find the value of an antiderivative with ease. Remember to always check if the derivative of the antiderivative is equal to the original function, and to use the correct technique for finding the antiderivative.