Substitution for Definite Integrals: Videos & Practice Problems

To evaluate definite integrals using substitution, two methods can be applied. The first method treats the integral as an indefinite integral, performing substitution and then applying the original bounds. The second method transforms the bounds according to the substitution before integrating. Both methods yield the same result, emphasizing the importance of correctly applying bounds. For example, for the integral from 0 to 2 of (x² + 1)³ * 2x dx, both methods lead to the final answer of 156, demonstrating the consistency of the substitution technique.

Definite Integrals

In evaluating definite integrals using substitution, we can apply similar techniques as with indefinite integrals, with the key difference being the presence of bounds. There are two effective methods to handle these bounds during the evaluation process.

Consider the definite integral from 0 to 2 of \( (x^2 + 1)^3 \cdot 2x \, dx \). To begin, we can treat this integral as if it were indefinite by ignoring the bounds temporarily. We select \( u = x^2 + 1 \), which leads to \( du = 2x \, dx \). This allows us to rewrite the integral as \( \int u^3 \, du \).

Next, we integrate with respect to \( u \). The antiderivative is calculated using the power rule, yielding \( \frac{1}{4} u^4 + C \). We then substitute back \( u = x^2 + 1 \) to get \( \frac{1}{4} (x^2 + 1)^4 + C \). However, since we are dealing with a definite integral, we must evaluate this expression at the bounds of 0 and 2.

Using the Fundamental Theorem of Calculus, we compute:

\[\frac{1}{4} \left( (2^2 + 1)^4 - (0^2 + 1)^4 \right) = \frac{1}{4} \left( 5^4 - 1^4 \right) = \frac{1}{4} (625 - 1) = \frac{624}{4} = 156.\]

For the second method, we will transform the bounds along with the integral. Again, we start with the same integral and substitution \( u = x^2 + 1 \) leading to \( du = 2x \, dx \). This time, we also change the bounds: when \( x = 0 \), \( u = 1 \) and when \( x = 2 \), \( u = 5 \). Thus, the integral becomes \( \int_1^5 u^3 \, du \).

We proceed to integrate \( u^3 \) to get \( \frac{1}{4} u^4 \) and evaluate it at the new bounds:

\[\frac{1}{4} \left( 5^4 - 1^4 \right) = \frac{1}{4} (625 - 1) = \frac{624}{4} = 156.\]

Both methods yield the same result of 156. The choice of method depends on personal preference; either approach is valid. The first method treats the integral as indefinite before applying the bounds, while the second method incorporates the bounds into the substitution process from the start. Mastery of both techniques will enhance your ability to evaluate definite integrals effectively.

Evaluate the definite integral.
$\int_0^1\frac{t}{\sqrt{t^2+1}}dt$

• A. 1.414
• B. 0.414
• C. 1
• D. 0.707

Answer: (B)

Evaluate the definite integral.
$\int_1^2\left(x-3\right)\left(x^2-6x\right)^2dx$

• A. -64.5
• B. 533
• C. -2.33
• D. -1.17

Answer: (A)

Evaluate the definite integral.
${\int }_0^{\ln 2}\frac{e^{5y}}{3+e^{5y}}dy$

• A. 0.139
• B. 0.434
• C. -0.139
• D. -0.675

Answer: (B)

Here’s what students ask on this topic:

What is the substitution method for evaluating definite integrals?

The substitution method for evaluating definite integrals involves two approaches. The first method treats the integral as an indefinite integral, performing substitution and then applying the original bounds. This involves choosing a substitution variable $u$, finding $du$, and rewriting the integral in terms of $u$. After integrating, the original bounds are applied to the antiderivative. The second method transforms the bounds according to the substitution before integrating. This means substituting the original bounds into the function of $x$ to find new bounds in terms of $u$. Both methods yield the same result, emphasizing the importance of correctly applying bounds.

How do you apply the fundamental theorem of calculus in substitution for definite integrals?

The fundamental theorem of calculus is applied in substitution for definite integrals by evaluating the antiderivative at the bounds. After performing substitution and integrating, the antiderivative is calculated at the upper and lower bounds. For example, if the antiderivative is $\frac{1}{4}{u}^4$, and the bounds are transformed to 1 and 5, the fundamental theorem of calculus is used to find the difference: $\frac{1}{4}{5}^4-1^4$. This yields the final result of the definite integral.

What are the steps to evaluate a definite integral using substitution?

To evaluate a definite integral using substitution, follow these steps: 1) Choose a substitution variable $u$ and find $du$. 2) Rewrite the integral in terms of $u$ and $du$. 3) Integrate with respect to $u$. 4) If using the first method, replace $u$ with the original function of $x$ and apply the original bounds. If using the second method, transform the bounds according to the substitution and evaluate the antiderivative at these new bounds. Both methods will yield the same result.

Why is it important to correctly apply bounds in substitution for definite integrals?

Correctly applying bounds in substitution for definite integrals is crucial because it ensures the accuracy of the final result. The bounds determine the interval over which the integral is evaluated, and any error in applying them can lead to incorrect answers. In the first method, the original bounds are applied after substitution and integration. In the second method, the bounds are transformed according to the substitution, and the antiderivative is evaluated at these new bounds. Both methods require careful attention to the bounds to maintain consistency and correctness in the evaluation process.

Can both methods of substitution for definite integrals yield the same result?

Yes, both methods of substitution for definite integrals can yield the same result. The first method involves treating the integral as an indefinite integral, performing substitution, and then applying the original bounds. The second method transforms the bounds according to the substitution before integrating. Despite the different approaches, both methods are mathematically equivalent and will lead to the same final answer. This consistency demonstrates the reliability of the substitution technique, provided the bounds are correctly applied in each method.