Math / Calculus

QuestionFind $\int(12 x+5)^{2} d x$ $\int(12 x+5)^{2} d x=$

Studdy Solution

STEP 1

1. We are dealing with an indefinite integral.
2. The integrand is a polynomial expression raised to a power.
3. We will use substitution to simplify the integration process.

STEP 2

1. Use substitution to simplify the integrand.
2. Integrate the simplified expression.
3. Substitute back to the original variable.

STEP 3

To simplify the integration process, use substitution. Let:
$u = 12x + 5$
Then, differentiate $u$ with respect to $x$ :
$\frac{du}{dx} = 12$
This implies:
$du = 12 \, dx$
Therefore, we can express $dx$ in terms of $du$ :
$dx = \frac{1}{12} \, du$
Substitute $u$ and $dx$ into the integral:
$\int (12x + 5)^2 \, dx = \int u^2 \cdot \frac{1}{12} \, du$
$= \frac{1}{12} \int u^2 \, du$

STEP 4

Now, integrate the simplified expression:
$\frac{1}{12} \int u^2 \, du$
The integral of $u^2$ with respect to $u$ is:
$\int u^2 \, du = \frac{u^3}{3} + C$
where $C$ is the constant of integration.
Thus, the integral becomes:
$\frac{1}{12} \cdot \left( \frac{u^3}{3} \right) + C$
$= \frac{1}{36} u^3 + C$

STEP 5

Substitute back the original expression for $u$ :
$u = 12x + 5$
So, the integral becomes:
$\frac{1}{36} (12x + 5)^3 + C$
The indefinite integral is:
$\boxed{\frac{1}{36} (12x + 5)^3 + C}$

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