Math  /  Calculus

QuestionSuppose we want to evaluate the definite integral, 2(5x+7)2dx\int_{2} \overline{(5 x+7)^{2}} d x using the substitution, u=5x+7u=5 x+7.
Part 1.
Re-write the definite integral in terms of the variable uu and remember to use the limits of integration for the function u=f(x)u=f(x). Then, inferut the antiderivative of the integrand and the limits of integration you found. \square ==\square \square =[]=[\square]
Part 2.

Studdy Solution

STEP 1

What is this asking? We're going to swap out xx with a new buddy uu in our integral, making it way easier to solve, then we'll actually solve it! Watch out! Don't forget to change the limits of integration when you switch to uu!
It's a classic trip-up.

STEP 2

1. Define the substitution
2. Transform the integral
3. Calculate the indefinite integral
4. Evaluate the definite integral

STEP 3

Let's **define** our substitution!
We're letting u=5x+7u = 5x + 7.

STEP 4

Now, we need to find dudu in terms of dxdx.
Taking the derivative of both sides with respect to xx, we get dudx=5\frac{du}{dx} = 5.

STEP 5

So, du=5dxdu = 5 \cdot dx.
This means dx=15dudx = \frac{1}{5} \cdot du.
We'll need this later!

STEP 6

Time to **transform** our integral!
We're given the integral 2(5x+7)2dx\int_{2} (5x+7)^2 dx.

STEP 7

Remember how we said u=5x+7u = 5x + 7?
Let's **substitute** that in!
We also found that dx=15dudx = \frac{1}{5} du, so let's plug that in too.

STEP 8

Now, about those limits of integration.
When x=2x = 2, u=52+7=17u = 5 \cdot 2 + 7 = 17.
So, our new lower limit is **17**.
Since the upper limit of integration is not given, we will calculate the indefinite integral.
Our transformed integral is u215du\int u^2 \cdot \frac{1}{5} du.
Much nicer, right?

STEP 9

Let's **calculate** that integral!
We can pull the constant 15\frac{1}{5} out front: 15u2du\frac{1}{5} \int u^2 du.

STEP 10

Using the power rule for integration, we add 1 to the exponent and divide by the new exponent: 15u2du=15u33+C\frac{1}{5} \int u^2 du = \frac{1}{5} \cdot \frac{u^3}{3} + C.

STEP 11

Simplifying, we get u315+C\frac{u^3}{15} + C.

STEP 12

We found the indefinite integral is u315+C\frac{u^3}{15} + C.
Since we are asked to calculate the indefinite integral, we can ignore the constant CC.

STEP 13

Substituting back u=5x+7u = 5x + 7, we get (5x+7)315\frac{(5x+7)^3}{15}.

STEP 14

The indefinite integral is (5x+7)315\frac{(5x+7)^3}{15}.

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