Problem 45

Question

Evaluate the integrals in Exercises $13-48$ . $$ \int t^{3}\left(1+t^{4}\right)^{3} d t $$

Step-by-Step Solution

Verified

Answer

$ \frac{(1 + t^4)^4}{16} + C $

1Step 1: Decide on the substitution

We notice that the integral contains a composite function within a power: $ (1 + t^4)^3 $. This suggests using substitution. Let's set $ u = 1 + t^4 $.

2Step 2: Differentiate the substitution

Now, we differentiate $ u $ with respect to $ t $. Since $ u = 1 + t^4 $, it follows that $ \frac{du}{dt} = 4t^3 $ or $ du = 4t^3 \, dt $. Hence, $ t^3 \, dt = \frac{1}{4} \, du $.

3Step 3: Substitute into the integral

Substitute $ u = 1 + t^4 $ and $ t^3 \, dt = \frac{1}{4} \, du $ into the integral. The integral becomes: \[ \int t^3 (1 + t^4)^3 \, dt = \int \frac{1}{4} u^3 \, du. \]

4Step 4: Solve the new integral

Now we integrate with respect to $ u $: \[ \int \frac{1}{4} u^3 \, du = \frac{1}{4} \cdot \frac{u^4}{4} + C = \frac{u^4}{16} + C. \]

5Step 5: Substitute back to the original variable

Substitute back $ u = 1 + t^4 $ into $ \frac{u^4}{16} + C $, giving: \[ \frac{(1 + t^4)^4}{16} + C. \]

Key Concepts

Substitution MethodDefinite and Indefinite IntegralsPolynomial Integration

Substitution Method

The substitution method in integration helps to simplify an integral by reducing it to a more basic form. This is often done when the integral contains a composite function, where a part of the expression is more manageable if considered as a single unit. In the given problem, the expression

$(1 + t^4)^3$

suggests that we use substitution by letting:

$u = 1 + t^4$

This choice is made because differentiating $u$ with respect to $t$ gives us a simple expression,

$du = 4t^3 \, dt$

which allows us to express $t^3 \, dt$ in terms of $du$.

By substituting these into the original integral, we transform a potentially complex problem into a basic polynomial integral with respect to $u$,

$\int \frac{1}{4} u^3 \, du$

This step significantly simplifies the integration process.

Definite and Indefinite Integrals

Integrals can be classified into two main types: definite and indefinite integrals. A definite integral is one that is evaluated over a specific interval, providing a numerical result, while an indefinite integral represents a family of functions.

In the given problem,

$\int t^3 (1 + t^4)^3 \, dt$

we are dealing with an indefinite integral as no limits of integration are provided. Indefinite integrals find the antiderivative or the general form of functions. The process involves integrating the expression symbolically and appending a constant $C$, representing the constant of integration, crucial when switching from a derivative back to the original function.

After substituting and simplifying to the integral in terms of $u$:

$\int \frac{1}{4} u^3 \, du$

we integrate and add $C$ to signify that this solution represents a general form.

Polynomial Integration

Polynomial integration involves integrating expressions made up of sums of powers of variables, a process that aligns closely with the fundamental principles of calculus.

Usually taking the form $x^n$, where $n$ is a constant.

The rule for integrating an expression like this is to increase the power by one and then divide by the new power. For the polynomial $u^3$, the integral becomes: