Problem 95 - Calculus Single Variable

Question

Determine the value of the upper limit of integration \(b\) for which a substitution converts the integral on the left to the integral on the right. \(\int_{0}^{b}\left(3 t^{2}+1\right) \sec ^{2}\left(t^{3}+t\right) \tan ^{2}\left(t^{3}+t\right) d t=\int_{0}^{1} u^{2} d u\)

Step-by-Step Solution

Answer

The value of \( b \) is 1.

Step 1: Identify the substitution

We need to make a substitution that will transform the left integral into the right integral. Notice the term \( (t^3+t) \) in the original integral. Let's set \( u = t^3 + t \). This substitution aims to simplify the integral and relate it to the second integral.

Step 2: Differentiate the substitution

To complete the substitution, we find \( du \) in terms of \( dt \). Differentiate \( u = t^3 + t \) with respect to \( t \): \( \frac{du}{dt} = 3t^2 + 1 \). Therefore, \( du = (3t^2 + 1) dt \).

Step 3: Transform the integrals

Substitute \( u = t^3 + t \) and \( du = (3t^2 + 1) dt \) into the original integral. The differential \( dt \) matches perfectly with \( du/(3t^2 + 1) \), so our integral becomes \( \int \sec^2(u) \tan^2(u) \, du \). This shows that all substitutions and transformations are consistent.

Step 4: Determine the new limits after substitution

Determine the limits of the integral when substituting \( u \) for \( t \). When \( t = 0 \), \( u = 0^3 + 0 = 0 \). For \( t = b \), we have \( u = b^3 + b \). But we need \( u = 1 \) for the upper limit of the right-side integral, resulting in the equation \( b^3 + b = 1 \).

Step 5: Solve for \( b \)

Solve \( b^3 + b = 1 \) for \( b \). Rearrange as \( b^3 + b - 1 = 0 \). By inspection or trial, discover that \( b = 1 \) is a solution (as \( 1^3 + 1 - 1 = 1 \)), hence \( b = 1 \). Therefore, \( b = 1 \) satisfies the equation, matching the given upper limit for \( u = 1 \) on the right integral.

Key Concepts

Definite Integrals

A definite integral is a key idea in calculus, used to calculate the area under a curve from one point to another. In our problem, we have two definite integrals with different limits of integration. Generally, definite integrals are written in the form: \[ \int_{a}^{b} f(x) \, dx \]where \(a\) and \(b\) are the lower and upper limits, and \(f(x)\) is the function being integrated.

• In the given problem, the left integral is from \(t = 0\) to \(t = b\).
• The right integral spans from \(u = 0\) to \(u = 1\).

The task is to convert the left integral to match the right one by finding the right value for the upper limit \(b\) on the left integral. This requires understanding and utilizing the concepts of substitution and limit transformation.

Substitution Method

The substitution method simplifies integration by introducing a new variable to replace a part of the expression. This technique is crucial for integrals that are too complex to handle directly. In our example, we use the following steps:

• Set up a substitution: Let \(u = t^3 + t\). This helps simplify the terms in the integrand.
• Differentiate to find \(du\): Differentiate \(u = t^3 + t\) to get \(\frac{du}{dt} = 3t^2 + 1\).
• Solve for \(dt\): You'll get \(du = (3t^2 + 1)\, dt\), matching terms with parts of the original integral.

The goal is to transform all the \(t\) terms into \(u\) terms so that the original integral becomes much simpler to evaluate. With this method, conversions between integrals become more straightforward.

Transforming Integrals

Transforming an integral involves changing its form without altering its value, often making it easier to evaluate. By using substitution, we effectively alter the limits and the integrand:

• Starting from our substitution \(u = t^3 + t\), we convert the limits by considering \(t=0\) gives \(u=0\) and \(t=b\) gives \(u=b^3 + b\).
• The target is to have \(u = 1\) for the upper limit. Solve \(b^3 + b = 1\) for \(b\).
• Testing values, we find \(b = 1\) satisfies \(b^3 + b = 1\), thus transforming the original integral's limits directly into the limits of 0 to 1 for the \(u\) integral.

The derivative of \(t^3 + t\) aligns with a part of the original integrand, allowing a smooth transition into the transformed integral \(\int_{0}^{1} u^2 \, du\). Hence, the original complex integral is reformed into a simpler equivalent, proving the limits and values are consistent.