Problem 35
Question
Use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{0}^{1}\left(x^{2}+1\right)^{10}(2 x) d x $$
Step-by-Step Solution
Verified Answer
\( \frac{2047}{11} \)
1Step 1: Identify the Substitution
In integrals like this, we use substitution to simplify the integral. Let's choose a substitution where the inside of the parentheses becomes our new variable. We set \( u = x^2 + 1 \).
2Step 2: Differentiate the Substitution
Differentiate the substitution equation \( u = x^2 + 1 \) with respect to \( x \). This gives the derivative \( du/dx = 2x \), so \( du = 2x \, dx \).
3Step 3: Adjust the Integral Limits
Since our substitution changes the variable of integration from \( x \) to \( u \), we need to change the limits of integration. When \( x = 0 \), \( u = 0^2 + 1 = 1 \). When \( x = 1 \), \( u = 1^2 + 1 = 2 \). So our new limits are from 1 to 2.
4Step 4: Substitute and Simplify the Integral
Substitute \( u \) and \( du \) into the integral. The integral becomes:\[ \int_{1}^{2} u^{10} \, du \].
5Step 5: Integrate
Evaluate the integral \( \int u^{10} \, du \). The antiderivative of \( u^{10} \) is \( \frac{u^{11}}{11} \).
6Step 6: Apply the Limits
Apply the limits to the antiderivative. This step gives:\[ \left[ \frac{u^{11}}{11} \right]_{1}^{2} = \frac{2^{11}}{11} - \frac{1^{11}}{11} \].
7Step 7: Simplify the Result
Calculate \( 2^{11} = 2048 \). Therefore, the evaluation becomes:\[ \frac{2048}{11} - \frac{1}{11} = \frac{2047}{11} \].
Key Concepts
Substitution RuleIntegral LimitsAntiderivatives
Substitution Rule
The substitution rule is a powerful tool for evaluating integrals, particularly definite integrals. It involves changing variables to make integration more manageable.
First, identify a part of the integral that can be replaced with a single variable. In our case, the expression \( (x^2 + 1) \) inside the integral became our new variable, which we named \( u \). This substitution simplifies the process by transforming the original integral into a form that might be easier to integrate.
After choosing the substitution, differentiate it to express \( dx \) in terms of \( du \). Here, differentiating \( u = x^2 + 1 \) leads us to \( du = 2x \, dx \). This step ensures that every part of the integrand is accounted for in the variable change.
First, identify a part of the integral that can be replaced with a single variable. In our case, the expression \( (x^2 + 1) \) inside the integral became our new variable, which we named \( u \). This substitution simplifies the process by transforming the original integral into a form that might be easier to integrate.
After choosing the substitution, differentiate it to express \( dx \) in terms of \( du \). Here, differentiating \( u = x^2 + 1 \) leads us to \( du = 2x \, dx \). This step ensures that every part of the integrand is accounted for in the variable change.
- Choose a substitution \( u \) to simplify \( f(x) \).
- Differential \( dx \) needs to be rewritten as \( du \).
- Re-express the entire integral in terms of \( u \) and \( du \).
Integral Limits
When applying the substitution rule to definite integrals, it is vital to adjust the integral limits to match the new variable.
Integral limits tell us the range over which to evaluate the integral. In our exercise, the original limits were \( x = 0 \) to \( x = 1 \). As \( x \) is converted to \( u \), these limits must be converted as well.
For \( x = 0 \), substitute into the equation \( u = x^2 + 1 \). This gives \( u = 1 \). Likewise, for \( x = 1 \), we get \( u = 2 \). Thus, the new limits are from \( u = 1 \) to \( u = 2 \).
Integral limits tell us the range over which to evaluate the integral. In our exercise, the original limits were \( x = 0 \) to \( x = 1 \). As \( x \) is converted to \( u \), these limits must be converted as well.
For \( x = 0 \), substitute into the equation \( u = x^2 + 1 \). This gives \( u = 1 \). Likewise, for \( x = 1 \), we get \( u = 2 \). Thus, the new limits are from \( u = 1 \) to \( u = 2 \).
- Substitute the original limits into the substitution equation.
- Find the new limits for the integral.
- Use these new limits in the transformed integral.
Antiderivatives
An antiderivative is a function whose derivative equates to the original function given in the integrand.
Identifying the antiderivative is essential in solving both indefinite and definite integrals. In our exercise, once the integral was transformed using substitution, we needed to find the antiderivative of \( u^{10} \).
The antiderivative of \( u^{10} \) is \( \frac{u^{11}}{11} \). With this antiderivative, we can then apply the fundamental theorem of calculus to evaluate the definite integral within the adjusted limits.
Identifying the antiderivative is essential in solving both indefinite and definite integrals. In our exercise, once the integral was transformed using substitution, we needed to find the antiderivative of \( u^{10} \).
The antiderivative of \( u^{10} \) is \( \frac{u^{11}}{11} \). With this antiderivative, we can then apply the fundamental theorem of calculus to evaluate the definite integral within the adjusted limits.
- Find the antiderivative of the given expression.
- Apply the antiderivative to the new limits defined by the substitution.
- Compute the result to evaluate the integral.
Other exercises in this chapter
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