QuestionEvaluate the integral: $\int \frac{(\ln x)^{2}}{x} \, dx$

Studdy Solution

STEP 1

Assumptions1. We are given the integral $\int \frac{(\ln x)^{}}{x} d x$ . We know the basic rules of integration3. We know the method of integration by parts, which is given by $\int u dv = uv - \int v du$

STEP 2

Let's use the method of integration by parts. We need to identify 'u' and 'dv' in our integral.Let's choose $u = (\ln x)^{2}$ and $dv = \frac{1}{x} dx$

STEP 3

Now, we need to find 'du' and 'v'.The derivative of 'u' with respect to 'x' gives 'du': $du =2 \ln x \cdot \frac{1}{x} dx =2 \frac{\ln x}{x} dx$ The integral of 'dv' gives 'v': $v = \int dv = \int \frac{1}{x} dx = \ln x$

STEP 4

Now, we substitute 'u', 'v', 'du', and 'dv' into the integration by parts formula $\int u dv = uv - \int v du$

STEP 5

Substitute the values for 'u', 'v', 'du', and 'dv' into the formula $\int \frac{(\ln x)^{2}}{x} dx = (\ln x)^{2} \cdot \ln x - \int \ln x \cdot2 \frac{\ln x}{x} dx$

STEP 6

implify the equation $= (\ln x)^{3} -2 \int \frac{(\ln x)^{2}}{x} dx$

STEP 7

Now, we see that the integral on the right side is the same as the original integral. Let's denote the original integral as '':
$= (\ln x)^{3} -2$

STEP 8

olve the equation for '':
$3 = (\ln x)^{3}$ $= \frac{1}{3}(\ln x)^{3}$ So, the solution to the integral $\int \frac{(\ln x)^{2}}{x} dx$ is $\frac{1}{3}(\ln x)^{3}$ .

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QuestionEvaluate the integral: $\int \frac{(\ln x)^{2}}{x} \, dx$

Studdy Solution

STEP 1

Assumptions1. We are given the integral $\int \frac{(\ln x)^{}}{x} d x$ . We know the basic rules of integration3. We know the method of integration by parts, which is given by $\int u dv = uv - \int v du$

STEP 2

Let's use the method of integration by parts. We need to identify 'u' and 'dv' in our integral.Let's choose $u = (\ln x)^{2}$ and $dv = \frac{1}{x} dx$

STEP 3

Now, we need to find 'du' and 'v'.The derivative of 'u' with respect to 'x' gives 'du': $du =2 \ln x \cdot \frac{1}{x} dx =2 \frac{\ln x}{x} dx$ The integral of 'dv' gives 'v': $v = \int dv = \int \frac{1}{x} dx = \ln x$

STEP 4

Now, we substitute 'u', 'v', 'du', and 'dv' into the integration by parts formula $\int u dv = uv - \int v du$

STEP 5

Substitute the values for 'u', 'v', 'du', and 'dv' into the formula $\int \frac{(\ln x)^{2}}{x} dx = (\ln x)^{2} \cdot \ln x - \int \ln x \cdot2 \frac{\ln x}{x} dx$

STEP 6

implify the equation $= (\ln x)^{3} -2 \int \frac{(\ln x)^{2}}{x} dx$

STEP 7

Now, we see that the integral on the right side is the same as the original integral. Let's denote the original integral as '':
$= (\ln x)^{3} -2$

STEP 8

olve the equation for '':
$3 = (\ln x)^{3}$ $= \frac{1}{3}(\ln x)^{3}$ So, the solution to the integral $\int \frac{(\ln x)^{2}}{x} dx$ is $\frac{1}{3}(\ln x)^{3}$ .

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QuestionEvaluate the integral: ∫(ln⁡x)2x dx\int \frac{(\ln x)^{2}}{x} \, dx∫x(lnx)2​dx

Studdy Solution

STEP 1

Assumptions1. We are given the integral ∫(ln⁡x)xdx\int \frac{(\ln x)^{}}{x} d x∫x(lnx)​dx . We know the basic rules of integration3. We know the method of integration by parts, which is given by ∫udv=uv−∫vdu\int u dv = uv - \int v du∫udv=uv−∫vdu

STEP 2

Let's use the method of integration by parts. We need to identify 'u' and 'dv' in our integral.Let's chooseu=(ln⁡x)2u = (\ln x)^{2}u=(lnx)2anddv=1xdxdv = \frac{1}{x} dxdv=x1​dx

STEP 3

STEP 4

Now, we substitute 'u', 'v', 'du', and 'dv' into the integration by parts formula∫udv=uv−∫vdu\int u dv = uv - \int v du∫udv=uv−∫vdu

STEP 5

Substitute the values for 'u', 'v', 'du', and 'dv' into the formula∫(ln⁡x)2xdx=(ln⁡x)2⋅ln⁡x−∫ln⁡x⋅2ln⁡xxdx\int \frac{(\ln x)^{2}}{x} dx = (\ln x)^{2} \cdot \ln x - \int \ln x \cdot2 \frac{\ln x}{x} dx∫x(lnx)2​dx=(lnx)2⋅lnx−∫lnx⋅2xlnx​dx

STEP 6

implify the equation=(ln⁡x)3−2∫(ln⁡x)2xdx= (\ln x)^{3} -2 \int \frac{(\ln x)^{2}}{x} dx=(lnx)3−2∫x(lnx)2​dx

STEP 7

Now, we see that the integral on the right side is the same as the original integral. Let's denote the original integral as '':=(ln⁡x)3−2 = (\ln x)^{3} -2=(lnx)3−2

STEP 8

olve the equation for '':3=(ln⁡x)33 = (\ln x)^{3}3=(lnx)3=13(ln⁡x)3 = \frac{1}{3}(\ln x)^{3}=31​(lnx)3So, the solution to the integral ∫(ln⁡x)2xdx\int \frac{(\ln x)^{2}}{x} dx∫x(lnx)2​dx is 13(ln⁡x)3\frac{1}{3}(\ln x)^{3}31​(lnx)3.

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QuestionEvaluate the integral: $\int \frac{(\ln x)^{2}}{x} \, dx$

Assumptions1. We are given the integral $\int \frac{(\ln x)^{}}{x} d x$ . We know the basic rules of integration3. We know the method of integration by parts, which is given by $\int u dv = uv - \int v du$

Let's use the method of integration by parts. We need to identify 'u' and 'dv' in our integral.Let's choose $u = (\ln x)^{2}$ and $dv = \frac{1}{x} dx$

Now, we substitute 'u', 'v', 'du', and 'dv' into the integration by parts formula $\int u dv = uv - \int v du$

Substitute the values for 'u', 'v', 'du', and 'dv' into the formula $\int \frac{(\ln x)^{2}}{x} dx = (\ln x)^{2} \cdot \ln x - \int \ln x \cdot2 \frac{\ln x}{x} dx$

implify the equation $= (\ln x)^{3} -2 \int \frac{(\ln x)^{2}}{x} dx$

Now, we see that the integral on the right side is the same as the original integral. Let's denote the original integral as '':
$= (\ln x)^{3} -2$

olve the equation for '':
$3 = (\ln x)^{3}$ $= \frac{1}{3}(\ln x)^{3}$ So, the solution to the integral $\int \frac{(\ln x)^{2}}{x} dx$ is $\frac{1}{3}(\ln x)^{3}$ .