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mronjo7n

Answered question

2021-11-21

Use a table of integrals with forms involving ln u to find the indefinite integral

\int x^{6} \ln x d x

Sue Leahy

Beginner2021-11-22Added 13 answers

Step 1
Given that,
Indefinite Integral

\int x^{6} \ln x d x

As we know

\int u d v = u v - \int v d u

Step 2
Here,

u = \ln (x)

v^{'} = x^{6}

\int x^{6} \ln (x) d x = \frac{1}{7} x^{7} \ln (x) - \int \frac{x^{6}}{7} d x

= \frac{1}{7} x^{7} \ln (x) - \frac{1}{7} \cdot \frac{x^{7}}{7}

= \frac{x^{7} \ln (x)}{7} - \frac{x^{7}}{49} + C

\int x^{6} \ln (x) d x = \frac{x^{7} \ln (x)}{7} - \frac{x^{7}}{49} + C

Ryan Willis

Beginner2021-11-23Added 15 answers

Step 1: Use Integration by Parts on

\int x^{6} \ln x d x

.
Let

u = \ln x, d v = x^{6}, d u = \frac{1}{x} d x, v = \frac{x^{7}}{7}

Step 2: Substitute the above into

u v - \int v d u

\frac{x^{7} \ln x}{7} - \int \frac{x^{6}}{7} d x

Step 3: Use Constant Factor Rule:

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

\frac{x^{7} \ln x}{7} - \frac{1}{7} \int x^{6} d x

Step 4: Use Power Rule:

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

\frac{x^{7} \ln x}{7} - \frac{x^{7}}{49}

Step 5: Add constant.

\frac{x^{7} \ln x}{7} - \frac{x^{7}}{49} + C

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