Use integration by parts to find the integral: ∫7xln⁡xdx

Step 1Consider the integral∫7xln⁡xdxStep 2Integration by parts ∫uv=u∫vdx−∫u′(∫vdx)dx∫7xln⁡xdxHereu=ln⁡xv=7xNow∫7xln⁡xdx=ln⁡x∫7xdx−∫(ln⁡x)′(∫7xdx)dx=ln⁡x[7x22]−∫(1x)[7x22]dx (∵(ln⁡x)′=1x)=7x22ln⁡x−72∫xdx=7x22ln⁡x−72[x22]=7x22ln⁡x−7x24∫7xln⁡xdx=7x22ln⁡x−7x24Step 3Answer:∫7xln⁡xdx=7x22ln⁡x−7x24

Step 1: Use Constant Factor Rule: ∫cf(x)dx=c∫f(x)dx. 7∫xln⁡xdx Step 2: Use Integration by Parts on ∫xln⁡xdx. Let u=ln⁡x,dv=x,du=1xdx,v=x22 Step 3: Substitute the above into uv−∫vdu. 7(x2ln⁡x2−∫x2dx) Step 4: Use Constant Factor Rule: ∫cf(x)dx=c∫f(x)dx. 7(x2ln⁡x2−12∫xdx) Step 5: Use Power Rule: ∫xndx=xn+1n+1+C. 7(x2ln⁡x2−x24) Step 6: Add constant. 7(x2ln⁡x2−x24)+C

Step 1Consider the integral∫7xln⁡xdxStep 2Integration by parts ∫uv=u∫vdx−∫u′(∫vdx)dx∫7xln⁡xdxHereu=ln⁡xv=7xNow∫7xln⁡xdx=ln⁡x∫7xdx−∫(ln⁡x)′(∫7xdx)dx=ln⁡x[7x22]−∫(1x)[7x22]dx (∵(ln⁡x)′=1x)=7x22ln⁡x−72∫xdx=7x22ln⁡x−72[x22]=7x22ln⁡x−7x24∫7xln⁡xdx=7x22ln⁡x−7x24Step 3Answer:∫7xln⁡xdx=7x22ln⁡x−7x24

Step 1: Use Constant Factor Rule: ∫cf(x)dx=c∫f(x)dx. 7∫xln⁡xdx Step 2: Use Integration by Parts on ∫xln⁡xdx. Let u=ln⁡x,dv=x,du=1xdx,v=x22 Step 3: Substitute the above into uv−∫vdu. 7(x2ln⁡x2−∫x2dx) Step 4: Use Constant Factor Rule: ∫cf(x)dx=c∫f(x)dx. 7(x2ln⁡x2−12∫xdx) Step 5: Use Power Rule: ∫xndx=xn+1n+1+C. 7(x2ln⁡x2−x24) Step 6: Add constant. 7(x2ln⁡x2−x24)+C

Solved: "Use integration by parts to find the integral:..."

elchatosarapage

Answered question

2021-11-19

Use integration by parts to find the integral:

\int 7 x \ln x d x

Answer & Explanation

Annie Midgett

Beginner2021-11-20Added 7 answers

Step 1
Consider the integral

\int 7 x \ln x d x

Step 2
Integration by parts

\int u v = u \int v d x - \int u^{'} (\int v d x) d x

\int 7 x \ln x d x

Here

u = \ln x

v=7x
Now

\int 7 x \ln x d x = \ln x \int 7 x d x - \int {(\ln x)}^{'} (\int 7 x d x) d x

= \ln x [\frac{7 x^{2}}{2}] - \int (\frac{1}{x}) [\frac{7 x^{2}}{2}] d x (∵ {(\ln x)}^{'} = \frac{1}{x})

= \frac{7 x^{2}}{2} \ln x - \frac{7}{2} \int x d x

= \frac{7 x^{2}}{2} \ln x - \frac{7}{2} [\frac{x^{2}}{2}]

= \frac{7 x^{2}}{2} \ln x - \frac{7 x^{2}}{4}

\int 7 x \ln x d x = \frac{7 x^{2}}{2} \ln x - \frac{7 x^{2}}{4}

Step 3
Answer:

\int 7 x \ln x d x = \frac{7 x^{2}}{2} \ln x - \frac{7 x^{2}}{4}

Sact1989

Beginner2021-11-21Added 10 answers

Step 1: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

7 \int x \ln x d x

Step 2: Use Integration by Parts on

\int x \ln x d x

.
Let

u = \ln x, d v = x, d u = \frac{1}{x} d x, v = \frac{x^{2}}{2}

Step 3: Substitute the above into

u v - \int v d u

7 (\frac{x^{2} \ln x}{2} - \int \frac{x}{2} d x)

Step 4: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

7 (\frac{x^{2} \ln x}{2} - \frac{1}{2} \int x d x)

Step 5: Use Power Rule:

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C

7 (\frac{x^{2} \ln x}{2} - \frac{x^{2}}{4})

Step 6: Add constant.

7 (\frac{x^{2} \ln x}{2} - \frac{x^{2}}{4}) + C

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