integral of (3p^2-2)^2e^{-4p} — step-by-step solution

1) Before performing the integration, let's calculate:

{(3 p^{2} - 2)}^{2} \cdot e^{(-4 p)}

\frac{{(3 p^{2} - 2)}^{2}}{e^{(4 p)}}

2) Let's integrate:

\int(\frac{{(3 p^{2} - 2)}^{2}}{e^{(4 p)}})dp

For

\frac{{(3 p^{2} - 2)}^{2}}{e^{(4 p)}}

, we use integration by parts:

\int(u1 \cdot v2')dp = u1 \cdot v2 - \int(u1' \cdot v2)dp

, with u1 =

{(3 p^{2} - 2)}^{2}

and v2' =

\frac{1}{e^{(4 p)}}

3) Let's differentiate u1:

\frac{d}{d p} ({(3 p^{2} - 2)}^{2})

Apply the differentiation rule:

\frac{d}{d p} (f^{n}) = n \cdot \frac{d}{d p} (f) \cdot f^{(n - 1)}

with f =

3 p^{2} - 2

2 \cdot \frac{d}{d p} (3 p^{2} - 2) \cdot (3 p^{2} - 2)

ℹ

2 \cdot \frac{d}{d p} (3 p^{2} - 2) \cdot (3 p^{2} - 2)

6 \cdot \frac{d}{d p} (3 p^{2} - 2) \cdot p^{2} - 4 \cdot \frac{d}{d p} (3 p^{2} - 2)

‖

\frac{d}{d p} (3 p^{2} - 2)

\frac{d}{d p} (3 p^{2}) + 0

6 \cdot (\frac{d}{d p} (3 p^{2}) + 0) \cdot p^{2} - 4 \cdot (\frac{d}{d p} (3 p^{2}) + 0)

ℹ

\frac{d}{d p} (3 p^{2})

3 \cdot \frac{d}{d p} (p^{2})

‖

\frac{d}{d p} (p^{2})

2 \cdot \frac{d}{d p} (p) \cdot p

6 \cdot 3 \cdot 2 \cdot \frac{d}{d p} (p) \cdot p \cdot p^{2} - 4 \cdot 3 \cdot 2 \cdot \frac{d}{d p} (p) \cdot p

ℹ

6 \cdot 3

18

‖

p p^{2}

p^{3}

‖

4 \cdot 3

12

‖

\frac{d}{d p} (p)

1

18 \cdot 2 \cdot \frac{d}{d p} (p) \cdot p^{3} - 12 \cdot 2 \cdot 1 \cdot p

ℹ

18 \cdot 2

36

‖

\frac{d}{d p} (p)

1

36 \cdot 1 \cdot p^{3} - 12 \cdot 2 p

36 p^{3} - 24 p

Thus,

u1'

36 p^{3} - 24 p

4) Let's compute the integral v2':

\int(\frac{1}{e^{(4 p)}})dp

We attempt to reverse an integration: starting from

\frac{1}{e^{(4 p)}}

and

\frac{d}{d x} (\frac{1}{a^{n}}) = - \frac{\frac{d}{d x} (n \cdot \ln (a))}{a^{n}}

, we differentiate using exp

5) Let's perform the differentiation:

\frac{d}{d p} (\frac{1}{e^{(4 p)}})

⇥We differentiate

u3

1

and

v4

e^{(4 p)}

using:

\frac{d}{d p} (\frac{u3}{v4}) = \frac{\frac{d}{d p} (u3) \cdot v4 - u3 \cdot \frac{d}{d p} (v4)}{{v4}^{2}}

⇥5.1) Let's differentiate

u3

\frac{d}{d p} (1)

⇥

0

⇥5.2) Let's differentiate

v4

\frac{d}{d p} (e^{(4 p)})

⇥We differentiate using:

\frac{d}{d p} (\exp (f)) = \frac{d}{d p} (f) \cdot \exp (f)

(where f =

4 p

)

⇥

\frac{d}{d p} (4 p) \cdot e^{(4 p)}

⇥

4 e^{(4 p)}

⇥5.3) You can evaluate the resulted derivative (

\frac{u3}{v4}

)':

\frac{0 \cdot e^{(4 p)} - 1 \cdot 4 e^{(4 p)}}{{(e^{(4 p)})}^{2}}

⇥ℹ

0 \cdot e^{(4 p)}

0

‖

{(e^{(4 p)})}^{2}

e^{(4 p \cdot 2)}

⇥

\frac{0 - 4 e^{(4 p)}}{e^{(4 p \cdot 2)}}

⇥

- \frac{4 e^{(4 p)}}{e^{(8 p)}}

⇥

-4 e^{(4 p - 8 p)}

⇥

- 4 e^{(- 4 p)}

- \frac{4}{e^{(4 p)}}

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

\frac{1}{-4} \cdot \frac{1}{e^{(4 p)}}

- \frac{1}{4} \cdot \frac{1}{e^{(4 p)}}

- \frac{1}{4 e^{(4 p)}}

So,

v2

corresponds to

- \frac{1}{4 e^{(4 p)}}

6) Let's calculate

u1' \cdot v2

(36 p^{3} - 24 p) \cdot (- \frac{1}{4 e^{(4 p)}})

ℹ

(36 p^{3} - 24 p) \cdot (- \frac{1}{4 e^{(4 p)}})

- \frac{36 p^{3} - 24 p}{4 e^{(4 p)}}

‖

- \frac{36 p^{3} - 24 p}{4 e^{(4 p)}}

- \frac{9 p^{3} - 6 p}{e^{(4 p)}}

- \frac{9 p^{3} - 6 p}{e^{(4 p)}}

\frac{- 9 p^{3} + 6 p}{e^{(4 p)}}

\frac{3 p \cdot (- 3 p^{2} + 2)}{e^{(4 p)}}

7) Let's compute the integral

u1 \cdot v2 - \int(u1' \cdot v2)dp

{(3 p^{2} - 2)}^{2} \cdot (- \frac{1}{4 e^{(4 p)}}) - \int(\frac{3 p \cdot (- 3 p^{2} + 2)}{e^{(4 p)}})dp

⇥For

\frac{3 p \cdot (- 3 p^{2} + 2)}{e^{(4 p)}}

, we use integration by parts:

\int(u3 \cdot v4')dp = u3 \cdot v4 - \int(u3' \cdot v4)dp

, with u3 =

p \cdot (- 3 p^{2} + 2)

and v4' =

\frac{1}{e^{(4 p)}}

⇥7.1) Let's differentiate u3:

\frac{d}{d p} (p \cdot (- 3 p^{2} + 2))

⇥⇥⇥For

U7

p

and

V8

- 3 p^{2} + 2

, use the product/quotient rule:

\frac{d}{d p} (U7 \cdot V8) = \frac{d}{d p} (U7) \cdot V8 + U7 \cdot \frac{d}{d p} (V8)

⇥⇥⇥Let's differentiate

U7

\frac{d}{d p} (p)

⇥⇥⇥

1

⇥⇥⇥Let's perform the differentiation

V8

\frac{d}{d p} (- 3 p^{2} + 2)

⇥⇥⇥

\frac{d}{d p} (- 3 p^{2}) + 0

⇥⇥⇥ℹ

\frac{d}{d p} (- 3 p^{2})

-3 \cdot \frac{d}{d p} (p^{2})

‖

\frac{d}{d p} (p^{2})

2 \cdot \frac{d}{d p} (p) \cdot p

⇥⇥⇥

-3 \cdot 2 \cdot \frac{d}{d p} (p) \cdot p

⇥⇥⇥ℹ

3 \cdot 2

6

‖

\frac{d}{d p} (p)

1

⇥⇥⇥

-6 \cdot 1 \cdot p

⇥⇥⇥

-6 p

⇥⇥⇥You can evaluate the resulted derivative (

U7 \cdot V8

)':

1 \cdot (- 3 p^{2} + 2) + p \cdot (- 6 p)

⇥⇥⇥

(- 3 p^{2} + 2) + p^{2} \cdot (-6)

⇥⇥⇥

- 3 p^{2} + 2 - p^{2} \cdot 6

⇥

- 9 p^{2} + 2

⇥So,

u3'

corresponds to

- 9 p^{2} + 2

⇥Thus,

v4

- \frac{1}{4 e^{(4 p)}}

⇥7.2) Let's evaluate

u3' \cdot v4

(- 9 p^{2} + 2) \cdot (- \frac{1}{4 e^{(4 p)}})

⇥

- \frac{- 9 p^{2} + 2}{4 e^{(4 p)}}

⇥

\frac{9 p^{2} - 2}{4 e^{(4 p)}}

⇥7.3) Let's compute the integral

u3 \cdot v4 - \int(u3' \cdot v4)dp

3 \cdot (p \cdot (- 3 p^{2} + 2) \cdot (- \frac{1}{4 e^{(4 p)}}) - \int(\frac{9 p^{2} - 2}{4 e^{(4 p)}})dp)

⇥⇥For

\frac{9 p^{2} - 2}{4 e^{(4 p)}}

, we use integration by parts:

\int(u5 \cdot v6')dp = u5 \cdot v6 - \int(u5' \cdot v6)dp

, with u5 =

9 p^{2} - 2

and v6' =

\frac{1}{e^{(4 p)}}

⇥⇥7.3.1) Let's differentiate u5:

\frac{d}{d p} (9 p^{2} - 2)

⇥⇥

\frac{d}{d p} (9 p^{2}) + 0

⇥⇥ℹ

\frac{d}{d p} (9 p^{2})

9 \cdot \frac{d}{d p} (p^{2})

‖

\frac{d}{d p} (p^{2})

2 \cdot \frac{d}{d p} (p) \cdot p

⇥⇥

9 \cdot 2 \cdot \frac{d}{d p} (p) \cdot p

⇥⇥ℹ

9 \cdot 2

18

‖

\frac{d}{d p} (p)

1

⇥⇥

18 \cdot 1 \cdot p

⇥⇥

18 p

⇥⇥So,

u5'

corresponds to

18 p

⇥⇥Thus,

v6

- \frac{1}{4 e^{(4 p)}}

⇥⇥7.3.2) Let's evaluate

u5' \cdot v6

18 p \cdot (- \frac{1}{4 e^{(4 p)}})

⇥⇥

- \frac{18 p}{4 e^{(4 p)}}

⇥⇥

- \frac{9 p}{2 e^{(4 p)}}

⇥⇥7.3.3) Let's integrate

u5 \cdot v6 - \int(u5' \cdot v6)dp

\frac{1}{4} \cdot ((9 p^{2} - 2) \cdot (- \frac{1}{4 e^{(4 p)}}) - \int(- \frac{9 p}{2 e^{(4 p)}})dp)

⇥⇥ℹ

(9 p^{2} - 2) \cdot (- \frac{1}{4 e^{(4 p)}})

- \frac{9 p^{2} - 2}{4 e^{(4 p)}}

‖

\frac{1}{4} \cdot (- \frac{9 p^{2} - 2}{4 e^{(4 p)}} - \int(- \frac{9 p}{2 e^{(4 p)}})dp)

\frac{- \frac{9 p^{2} - 2}{4 e^{(4 p)}} - \int(- \frac{9 p}{2 e^{(4 p)}})dp}{4}

⇥⇥

- \frac{\frac{9 p^{2} - 2}{4 e^{(4 p)}} + \int(- \frac{9 p}{2 e^{(4 p)}})dp}{4}

⇥⇥

- \frac{\frac{(9 p^{2} - 2) + \int(- \frac{9 p}{2 e^{(4 p)}})dp \cdot 4 e^{(4 p)}}{4 e^{(4 p)}}}{4}

⇥⇥⇥For

- \frac{9 p}{2 e^{(4 p)}}

, we use integration by parts:

\int(u7 \cdot v8')dp = u7 \cdot v8 - \int(u7' \cdot v8)dp

, with u7 =

p

and v8' =

\frac{1}{e^{(4 p)}}

⇥⇥⇥Thus,

u7'

1

⇥⇥⇥So

v8

is equal to

- \frac{1}{4 e^{(4 p)}}

⇥⇥⇥Let's compute the integral

u7 \cdot v8 - \int(u7' \cdot v8)dp

- \frac{9}{2} \cdot (p \cdot (- \frac{1}{4 e^{(4 p)}}) - \int(- \frac{1}{4 e^{(4 p)}})dp)

⇥⇥⇥For the math expression:

\frac{1}{4 e^{(4 p)}}

, we know the derivative formula:

\frac{d}{d x} (\frac{1}{a^{n}}) = - \frac{\frac{d}{d x} (n \cdot \ln (a))}{a^{n}}

, so we try differentiating with exp

⇥⇥⇥Let's perform the differentiation:

\frac{d}{d p} (\frac{1}{e^{(4 p)}})

⇥⇥⇥For

u13

1

and

v14

e^{(4 p)}

, use the product/quotient rule:

\frac{d}{d p} (\frac{u13}{v14}) = \frac{\frac{d}{d p} (u13) \cdot v14 - u13 \cdot \frac{d}{d p} (v14)}{{v14}^{2}}

⇥⇥⇥Let's perform the differentiation

u13

\frac{d}{d p} (1)

⇥⇥⇥

0

⇥⇥⇥Let's differentiate

v14

\frac{d}{d p} (e^{(4 p)})

⇥⇥⇥Apply the differentiation rule:

\frac{d}{d p} (\exp (f)) = \frac{d}{d p} (f) \cdot \exp (f)

with f =

4 p

⇥⇥⇥

\frac{d}{d p} (4 p) \cdot e^{(4 p)}

⇥⇥⇥

4 e^{(4 p)}

⇥⇥⇥You can evaluate the resulted derivative (

\frac{u13}{v14}

)':

\frac{0 \cdot e^{(4 p)} - 1 \cdot 4 e^{(4 p)}}{{(e^{(4 p)})}^{2}}

⇥⇥⇥ℹ

0 \cdot e^{(4 p)}

0

‖

{(e^{(4 p)})}^{2}

e^{(4 p \cdot 2)}

⇥⇥⇥

\frac{0 - 4 e^{(4 p)}}{e^{(4 p \cdot 2)}}

⇥⇥⇥

- \frac{4 e^{(4 p)}}{e^{(8 p)}}

⇥⇥⇥

-4 e^{(4 p - 8 p)}

⇥⇥⇥

- 4 e^{(- 4 p)}

⇥⇥⇥

- \frac{4}{e^{(4 p)}}

⇥⇥⇥You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

⇥⇥⇥

- \frac{9}{2} \cdot (- \frac{p}{4 e^{(4 p)}} + \frac{1}{-16} \cdot \frac{1}{e^{(4 p)}})

⇥⇥⇥

- \frac{9 \cdot (- (\frac{p}{4 e^{(4 p)}} + \frac{1}{16} \cdot \frac{1}{e^{(4 p)}}))}{2}

⇥⇥⇥

- \frac{- 9 \cdot (\frac{p}{4 e^{(4 p)}} + \frac{1}{16 e^{(4 p)}})}{2}

⇥⇥⇥

\frac{9 \cdot (\frac{p}{4 e^{(4 p)}} + \frac{1}{16 e^{(4 p)}})}{2}

⇥⇥⇥

\frac{9 \cdot (\frac{p \cdot 16 + 4}{4 e^{(4 p)} \cdot 16})}{2}

⇥⇥⇥

\frac{9 \cdot \frac{p \cdot 16 + 4}{64 e^{(4 p)}}}{2}

⇥⇥⇥ℹ

9 \cdot \frac{p \cdot 16 + 4}{64 e^{(4 p)}}

\frac{9 \cdot (p \cdot 16 + 4)}{64 e^{(4 p)}}

‖

\frac{9 \cdot (p \cdot 16 + 4)}{64 e^{(4 p)}}

\frac{9 \cdot (4 p + 1)}{16 e^{(4 p)}}

⇥⇥⇥

\frac{\frac{9 \cdot (4 p + 1)}{16 e^{(4 p)}}}{2}

⇥⇥⇥

\frac{9 \cdot (4 p + 1)}{16 e^{(4 p)}} \cdot \frac{1}{2}

⇥⇥⇥

\frac{9 \cdot (4 p + 1)}{32 e^{(4 p)}}

⇥⇥You can then resume the integration:

⇥⇥

- \frac{\frac{9 p^{2} - 2 + \frac{9 \cdot (4 p + 1)}{32 e^{(4 p)}} \cdot 4 e^{(4 p)}}{4 e^{(4 p)}}}{4}

⇥⇥ℹ

\frac{9 \cdot (4 p + 1)}{32 e^{(4 p)}} \cdot 4 e^{(4 p)}

\frac{36 \cdot (4 p + 1) \cdot e^{(4 p)}}{32 e^{(4 p)}}

‖

9 p^{2} + \frac{36 \cdot (4 p + 1) \cdot e^{(4 p)}}{32 e^{(4 p)}}

\frac{9 p^{2} \cdot 32 e^{(4 p)} + 36 \cdot (4 p + 1) \cdot e^{(4 p)}}{32 e^{(4 p)}}

⇥⇥

- \frac{\frac{\frac{9 p^{2} \cdot 32 e^{(4 p)} + 36 \cdot (4 p + 1) \cdot e^{(4 p)}}{32 e^{(4 p)}} - 2}{4 e^{(4 p)}}}{4}

⇥⇥ℹ

9 \cdot 32

288

‖

\frac{288 p^{2} e^{(4 p)} + 36 \cdot (4 p + 1) \cdot e^{(4 p)}}{32 e^{(4 p)}} - 2

\frac{(288 p^{2} e^{(4 p)} + 36 \cdot (4 p + 1) \cdot e^{(4 p)}) - 2 \cdot 32 e^{(4 p)}}{32 e^{(4 p)}}

⇥⇥

- \frac{\frac{\frac{(288 p^{2} e^{(4 p)} + 36 \cdot (4 p + 1) \cdot e^{(4 p)}) - 2 \cdot 32 e^{(4 p)}}{32 e^{(4 p)}}}{4 e^{(4 p)}}}{4}

⇥⇥

- \frac{\frac{\frac{288 p^{2} e^{(4 p)} + 36 \cdot (4 p + 1) \cdot e^{(4 p)} - 64 e^{(4 p)}}{32 e^{(4 p)}}}{4 e^{(4 p)}}}{4}

⇥⇥

- \frac{\frac{\frac{72 p^{2} e^{(4 p)} + 9 \cdot (4 p + 1) \cdot e^{(4 p)} - 16 e^{(4 p)}}{8 e^{(4 p)}}}{4 e^{(4 p)}}}{4}

⇥⇥

- \frac{e^{(4 p)} \cdot (72 p^{2} + 9 \cdot (4 p + 1) - 16)}{8 e^{(4 p)}} \cdot \frac{1}{4} \cdot \frac{1}{e^{(4 p)}} \cdot \frac{1}{4}

⇥⇥ℹ

\frac{1}{4} \cdot \frac{1}{4}

\frac{1}{16}

‖

\frac{e^{(4 p)} \cdot (72 p^{2} + 9 \cdot (4 p + 1) - 16)}{8 e^{(4 p)}}

\frac{72 p^{2} + 9 \cdot (4 p + 1) - 16}{8}

⇥⇥

- \frac{72 p^{2} + 9 \cdot (4 p + 1) - 16}{8} \cdot \frac{1}{16} \cdot \frac{1}{e^{(4 p)}}

⇥⇥

- \frac{72 p^{2} + 9 \cdot (4 p + 1) - 16}{128 e^{(4 p)}}

⇥⇥

- \frac{72 p^{2} + (36 p + 9) - 16}{128 e^{(4 p)}}

⇥⇥

- \frac{72 p^{2} + 36 p - 7}{128 e^{(4 p)}}

⇥Then you can continue the integral computation:

⇥

3 \cdot (- \frac{p \cdot (- 3 p^{2} + 2)}{4 e^{(4 p)}} + \frac{72 p^{2} + 36 p - 7}{128 e^{(4 p)}})

⇥

3 \cdot (\frac{- p \cdot (- 3 p^{2} + 2) \cdot 128 + (72 p^{2} + 36 p - 7) \cdot 4}{4 e^{(4 p)} \cdot 128})

⇥

3 \cdot \frac{- p \cdot (- 3 p^{2} + 2) \cdot 128 + (72 p^{2} + 36 p - 7) \cdot 4}{512 e^{(4 p)}}

⇥ℹ

3 \cdot \frac{- p \cdot (- 3 p^{2} + 2) \cdot 128 + (72 p^{2} + 36 p - 7) \cdot 4}{512 e^{(4 p)}}

\frac{3 \cdot (- p \cdot (- 3 p^{2} + 2) \cdot 128 + (72 p^{2} + 36 p - 7) \cdot 4)}{512 e^{(4 p)}}

‖

\frac{3 \cdot (- p \cdot (- 3 p^{2} + 2) \cdot 128 + (72 p^{2} + 36 p - 7) \cdot 4)}{512 e^{(4 p)}}

\frac{3 \cdot (- 32 p \cdot (- 3 p^{2} + 2) + (72 p^{2} + 36 p - 7))}{128 e^{(4 p)}}

⇥

\frac{3 \cdot (- 32 p \cdot (- 3 p^{2} + 2) + (72 p^{2} + 36 p - 7))}{128 e^{(4 p)}}

⇥

\frac{3 \cdot (- 32 p \cdot (- 3 p^{2} + 2) + 72 p^{2} + 36 p - 7)}{128 e^{(4 p)}}

⇥

\frac{3 \cdot (- (- 96 p^{3} + 64 p) + 72 p^{2} + 36 p - 7)}{128 e^{(4 p)}}

⇥

\frac{3 \cdot (96 p^{3} - 64 p + 72 p^{2} + 36 p - 7)}{128 e^{(4 p)}}

⇥

\frac{3 \cdot (96 p^{3} - 28 p + 72 p^{2} - 7)}{128 e^{(4 p)}}

Finally you can finish the integral computation:

- \frac{{(3 p^{2} - 2)}^{2}}{4 e^{(4 p)}} - \frac{3 \cdot (96 p^{3} - 28 p + 72 p^{2} - 7)}{128 e^{(4 p)}}

\frac{- {(3 p^{2} - 2)}^{2} \cdot 128 - 3 \cdot (96 p^{3} - 28 p + 72 p^{2} - 7) \cdot 4}{4 e^{(4 p)} \cdot 128}

ℹ

3 \cdot 4

12

‖

4 \cdot 128

512

- {(3 p^{2} - 2)}^{2} \cdot 128 + 12 \cdot (96 p^{3}

- 28 p

+ 72 p^{2}

- 7) 512 e^{(4 p)}

- \frac{32 \cdot {(3 p^{2} - 2)}^{2} + 3 \cdot (96 p^{3} - 28 p + 72 p^{2} - 7)}{128 e^{(4 p)}}

- \frac{32 \cdot {(3 p^{2} - 2)}^{2} + (288 p^{3} - 84 p + 216 p^{2} - 21)}{128 e^{(4 p)}}

- \frac{32 \cdot {(3 p^{2} - 2)}^{2} + 288 p^{3} - 84 p + 216 p^{2} - 21}{128 e^{(4 p)}}

Indefinite integrals are defined up to an additive constant C, so this yields:

- \frac{32 \cdot {(3 p^{2} - 2)}^{2} + 288 p^{3} - 84 p + 216 p^{2} - 21}{128 e^{(4 p)}} + C

▶Antiderivative:

- \frac{288 p^{3} + 216 p^{2} - 84 p + 32 \cdot {(3 p^{2} - 2)}^{2} - 21}{128 e^{(4 p)}} + C

, with C being any constant