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8_murik_8 [283]
3 years ago
9

Integrate the following w.r.t x 1) 2x^2/3. 2) (5-x)^23​

Mathematics
1 answer:
Kryger [21]3 years ago
8 0

Answer:

A) \int\frac{2x^2}{3}dx=\frac{2x^3}{9}+C

B) \int(5-x)^{23}dx=-\frac{(5-x)^{24}}{24}+C

Step-by-step explanation:

A)

So we have the integral:

\int\frac{2x^2}{3}dx

First, remove the constant multiple:

=\frac{2}{3}\int x^2\dx

Use the power rule, where:

\int x^ndx=\frac{x^{n+1}}{x+1}

Therefore:

\frac{2}{3}\int x^2\dx\\=\frac{2}{3}(\frac{x^{2+1}}{2+1})

Simplify:

=\frac{2}{3}(\frac{x^{3}}{3})

And multiply:

=\frac{2x^3}{9}

And, finally, plus C:

=\frac{2x^3}{9}+C

B)

We have the integral:

\int(5-x)^{23}dx

To solve, we can use u-substitute.

Let u equal 5-x. Then:

u=5-x\\du=-1dx

So:

\int(5-x)^{23}dx\\=\int-u^{23}du

Move the negative outside:

=-\int u^{23}du

Power rule:

=-(\frac{u^{23+1}}{23+1})

Add:

=-(\frac{u^{24}}{24})

Substitute back 5-x:

=-(\frac{(5-x)^{24}}{24})

Constant of integration:

=-\frac{(5-x)^{24}}{24}+C

And we're done!

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