How to find the anti-derivative of 3 x 3 ?

Question

How to find the anti-derivative of             3              x        3            ?

Amelia Walsh · Accepted Answer

∫                  3                              x          3                            d      x      First take out the constant &quot;&quot;3&quot;&quot;      3    ∫                  1                              x          3                            d      x      Apply property                     1                              x          3                      =          x              -        3                  3    ∫          x              -        3                    d      x            3                            x                      -            3            +            1                                      -        3        +        1             =      3          (                                    x                          -              2                                                -          2                    )      =      3          (      -                        1                          2                      x            2                              )      =Answer:      -                  3                    2                  x          2                      +    C

Armani Vang · Accepted Answer

In fact, the antiderivative and integral of a function are the same thing.
So, the antiderivative of

\frac{3}{x^{3}}

is the same as

\int \frac{3}{x^{3}} d x

.
And here we go,

\int \frac{3}{x^{3}} d x

We take the constant out, and we get,

= 3 \int \frac{1}{x^{3}} d x

Using a little bit of algebraic manipulation, we get

= 3 \int {(x^{3})}^{- 1} d x

= 3 \int x^{- 3} d x

Now, we use the power rule, which states that

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

, and so we get

= 3 \cdot \frac{x^{- 3 + 1}}{- 3 + 1}

= 3 \cdot \frac{x^{- 2}}{- 2}

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

We also know that

x^{- a} = \frac{1}{x^{a}}

.

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

= \frac{1.5}{x^{2}}

How to find the anti-derivative of 3/x^3?

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