Integrate Using u-Substitution integral of (x^3+1)^2 with respect to x

Solution:

Step:1

We recognize that u-substitution is not suitable for this integral. An alternative approach will be used.

Step:2

First, we expand the expression $(x^3 + 1)^2$.

Step:2.1

We rewrite $(x^3 + 1)^2$ as $(x^3 + 1)(x^3 + 1)$ and consider the integral $\int (x^3 + 1)(x^3 + 1) dx$.

Step:2.2

Next, we apply the distributive property to the integrand: $\int (x^3(x^3 + 1) + 1(x^3 + 1)) dx$.

Step:2.3

We distribute $x^3$ across the sum: $\int (x^3 \cdot x^3 + x^3 \cdot 1 + 1(x^3 + 1)) dx$.

Step:2.4

We continue distributing: $\int (x^3 \cdot x^3 + x^3 \cdot 1 + 1 \cdot x^3 + 1 \cdot 1) dx$.

Step:2.5

We reorder terms if necessary: $\int (x^3 \cdot x^3 + 1 \cdot x^3 + 1 \cdot x^3 + 1 \cdot 1) dx$.

Step:2.6

We combine like terms using the power rule $a^m \cdot a^n = a^{m+n}$: $\int (x^{3+3} + 1 \cdot x^3 + 1 \cdot x^3 + 1 \cdot 1) dx$.

Step:2.7

We add the exponents where applicable: $\int (x^6 + 1 \cdot x^3 + 1 \cdot x^3 + 1 \cdot 1) dx$.

Step:2.8

We multiply $x^3$ by 1: $\int (x^6 + x^3 + 1 \cdot x^3 + 1 \cdot 1) dx$.

Step:2.9

We multiply $x^3$ by 1 again: $\int (x^6 + x^3 + x^3 + 1 \cdot 1) dx$.

Step:2.10

We multiply 1 by 1: $\int (x^6 + x^3 + x^3 + 1) dx$.

Step:2.11

We combine like terms: $\int (x^6 + 2x^3 + 1) dx$.

Step:3

We split the integral into separate integrals: $\int x^6 dx + \int 2x^3 dx + \int 1 dx$.

Step:4

We apply the power rule to integrate $x^6$: $\frac{1}{7}x^7 + C + \int 2x^3 dx + \int 1 dx$.

Step:5

We factor out the constant 2 from the integral: $\frac{1}{7}x^7 + C + 2\int x^3 dx + \int 1 dx$.

Step:6

We integrate $x^3$ using the power rule: $\frac{1}{7}x^7 + C + 2\left(\frac{1}{4}x^4 + C\right) + \int 1 dx$.

Step:7

We apply the constant rule to integrate 1: $\frac{1}{7}x^7 + C + 2\left(\frac{1}{4}x^4 + C\right) + x + C$.

Step:8

We simplify the expression.

Step:8.1

We combine $\frac{1}{4}$ and $x^4$: $\frac{1}{7}x^7 + C + 2\left(\frac{x^4}{4} + C\right) + x + C$.

Step:8.2

We simplify the expression further: $\frac{x^7}{7} + \frac{x^4}{2} + x + C$.

Step:8.3

We reorder the terms if necessary: $\frac{1}{7}x^7 + \frac{1}{2}x^4 + x + C$.

Knowledge Notes:

The problem-solving process involves expanding a binomial expression and integrating term by term. Here are the relevant knowledge points:

1. Distributive Property: This property allows us to multiply a sum by a term by multiplying each addend of the sum by the term separately.

2. Power Rule for Exponents: When multiplying like bases, we add the exponents: $a^m \cdot a^n = a^{m+n}$.

3. Power Rule for Integration: The integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}$, provided $n \neq -1$.

4. Constant Multiple Rule: A constant can be factored out of an integral: $\int k \cdot f(x) dx = k \cdot \int f(x) dx$.

5. Constant Rule of Integration: The integral of a constant is equal to the constant multiplied by the variable of integration: $\int a dx = ax + C$.

6. Combining Like Terms: Terms that are the same can be added or subtracted from each other.

7. Simplification: After integrating, terms are combined and simplified to reach the final answer.