Math  /  Calculus

QuestionThe following integral will be evaluated using integration by parts with the indicated choices of uu and dvd v. xe6xdx;u=x,dv=e6xdx\int x e^{6 x} d x ; \quad u=x, d v=e^{6 x} d x
Determine dud u and vv. du=d u= \square v=v=\square
Evaluate the integral. (Remember the constant of integration.) \square

Studdy Solution

STEP 1

1. We are using the method of integration by parts, which is based on the formula: $ \int u \, dv = uv - \int v \, du \]
2. The choices for \( u \) and \( dv \) are given as \( u = x \) and \( dv = e^{6x} \, dx \).

STEP 2

1. Determine du du and v v .
2. Apply the integration by parts formula.
3. Evaluate the resulting integral.
4. Include the constant of integration.

STEP 3

Differentiate u=x u = x to find du du :
u=x u = x du=ddx(x)dx=1dx=dx du = \frac{d}{dx}(x) \, dx = 1 \, dx = dx
Integrate dv=e6xdx dv = e^{6x} \, dx to find v v :
dv=e6xdx dv = e^{6x} \, dx To integrate, we use the formula for the integral of an exponential function: v=e6xdx=16e6x+C v = \int e^{6x} \, dx = \frac{1}{6} e^{6x} + C Since we are finding the antiderivative for the purpose of integration by parts, we can ignore the constant C C here: v=16e6x v = \frac{1}{6} e^{6x}

STEP 4

Apply the integration by parts formula:
Given: udv=uvvdu \int u \, dv = uv - \int v \, du
Substitute the values: u=x,dv=e6xdx,du=dx,v=16e6x u = x, \quad dv = e^{6x} \, dx, \quad du = dx, \quad v = \frac{1}{6} e^{6x}
xe6xdx=x(16e6x)(16e6x)dx \int x e^{6x} \, dx = x \left(\frac{1}{6} e^{6x}\right) - \int \left(\frac{1}{6} e^{6x}\right) \, dx
Simplify: =x6e6x16e6xdx = \frac{x}{6} e^{6x} - \frac{1}{6} \int e^{6x} \, dx

STEP 5

Evaluate the remaining integral:
e6xdx=16e6x \int e^{6x} \, dx = \frac{1}{6} e^{6x}
Substitute back: x6e6x16(16e6x) \frac{x}{6} e^{6x} - \frac{1}{6} \left(\frac{1}{6} e^{6x}\right)
Simplify: =x6e6x136e6x = \frac{x}{6} e^{6x} - \frac{1}{36} e^{6x}

STEP 6

Include the constant of integration:
xe6xdx=x6e6x136e6x+C \int x e^{6x} \, dx = \frac{x}{6} e^{6x} - \frac{1}{36} e^{6x} + C
The evaluated integral is:
x6e6x136e6x+C \boxed{\frac{x}{6} e^{6x} - \frac{1}{36} e^{6x} + C}

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