Math

QuestionFind the integral of the function xx+2x \sqrt{x+2} with respect to xx: xx+2dx\int x \sqrt{x+2} d x.

Studdy Solution

STEP 1

Assumptions1. We are given the integral xx+dx\int x \sqrt{x+} d x . We are to solve this integral using the method of substitution

STEP 2

We start by choosing a suitable substitution. In this case, let's let u=x+2u = x +2. This simplifies the square root term in the integral.
u=x+2u = x +2

STEP 3

We differentiate uu with respect to xx to get du/dxdu/dx.
dudx=1\frac{du}{dx} =1

STEP 4

We rearrange the above equation to get dxdx in terms of dudu.
dx=dudx = du

STEP 5

Substitute uu and dxdx into the original integral. The integral becomes (u2)udu\int (u-2) \sqrt{u} du.
(u2)udu\int (u-2) \sqrt{u} du

STEP 6

We can break this integral into two simpler integrals.
uudu2udu\int u \sqrt{u} du - \int2 \sqrt{u} du

STEP 7

We can rewrite uuu \sqrt{u} as u3/2u^{3/2} and 2u2 \sqrt{u} as 2u1/22u^{1/2} to make the integrals easier to solve.
u3/2du2u1/2du\int u^{3/2} du - \int2u^{1/2} du

STEP 8

Now we can use the power rule for integration, which states that xndx=1n+1xn+1\int x^n dx = \frac{1}{n+1}x^{n+1}, to solve these integrals.
25u5/243u3/2+C\frac{2}{5}u^{5/2} - \frac{4}{3}u^{3/2} + C

STEP 9

Finally, we substitute x+2x +2 back in for uu to get the solution in terms of xx.
25(x+2)5/243(x+2)3/2+C\frac{2}{5}(x+2)^{5/2} - \frac{4}{3}(x+2)^{3/2} + CThis is the solution to the integral.

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