Cálculo Ejemplos

${(3 e^{4 x} + \cos^{2} (5 x))}^{6}$

Paso 1

Usa el teorema del binomio.

${(3 e^{4 x})}^{6} + 6 {(3 e^{4 x})}^{5} \cos^{2} (5 x) + 15 {(3 e^{4 x})}^{4} {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2

Simplifica cada término.

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Paso 2.1

Aplica la regla del producto a $3 e^{4 x}$ .

$3^{6} {(e^{4 x})}^{6} + 6 {(3 e^{4 x})}^{5} \cos^{2} (5 x) + 15 {(3 e^{4 x})}^{4} {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.2

Eleva $3$ a la potencia de $6$ .

$729 {(e^{4 x})}^{6} + 6 {(3 e^{4 x})}^{5} \cos^{2} (5 x) + 15 {(3 e^{4 x})}^{4} {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.3

Multiplica los exponentes en ${(e^{4 x})}^{6}$ .

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Paso 2.3.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$729 e^{4 x \cdot 6} + 6 {(3 e^{4 x})}^{5} \cos^{2} (5 x) + 15 {(3 e^{4 x})}^{4} {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.3.2

Multiplica $6$ por $4$ .

$729 e^{24 x} + 6 {(3 e^{4 x})}^{5} \cos^{2} (5 x) + 15 {(3 e^{4 x})}^{4} {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.4

Aplica la regla del producto a $3 e^{4 x}$ .

$729 e^{24 x} + 6 (3^{5} {(e^{4 x})}^{5}) \cos^{2} (5 x) + 15 {(3 e^{4 x})}^{4} {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.5

Eleva $3$ a la potencia de $5$ .

$729 e^{24 x} + 6 (243 {(e^{4 x})}^{5}) \cos^{2} (5 x) + 15 {(3 e^{4 x})}^{4} {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.6

Multiplica los exponentes en ${(e^{4 x})}^{5}$ .

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Paso 2.6.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$729 e^{24 x} + 6 (243 e^{4 x \cdot 5}) \cos^{2} (5 x) + 15 {(3 e^{4 x})}^{4} {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.6.2

Multiplica $5$ por $4$ .

$729 e^{24 x} + 6 (243 e^{20 x}) \cos^{2} (5 x) + 15 {(3 e^{4 x})}^{4} {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.7

Multiplica $243$ por $6$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 15 {(3 e^{4 x})}^{4} {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.8

Aplica la regla del producto a $3 e^{4 x}$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 15 (3^{4} {(e^{4 x})}^{4}) {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.9

Eleva $3$ a la potencia de $4$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 15 (81 {(e^{4 x})}^{4}) {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.10

Multiplica los exponentes en ${(e^{4 x})}^{4}$ .

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Paso 2.10.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 15 (81 e^{4 x \cdot 4}) {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.10.2

Multiplica $4$ por $4$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 15 (81 e^{16 x}) {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.11

Multiplica $81$ por $15$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} {(\cos^{2} (5 x))}^{2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.12

Multiplica los exponentes en ${(\cos^{2} (5 x))}^{2}$ .

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Paso 2.12.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} {\cos (5 x)}^{2 \cdot 2} + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.12.2

Multiplica $2$ por $2$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 20 {(3 e^{4 x})}^{3} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.13

Aplica la regla del producto a $3 e^{4 x}$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 20 (3^{3} {(e^{4 x})}^{3}) {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.14

Eleva $3$ a la potencia de $3$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 20 (27 {(e^{4 x})}^{3}) {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.15

Multiplica los exponentes en ${(e^{4 x})}^{3}$ .

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Paso 2.15.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 20 (27 e^{4 x \cdot 3}) {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.15.2

Multiplica $3$ por $4$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 20 (27 e^{12 x}) {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.16

Multiplica $27$ por $20$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} {(\cos^{2} (5 x))}^{3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.17

Multiplica los exponentes en ${(\cos^{2} (5 x))}^{3}$ .

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Paso 2.17.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} {\cos (5 x)}^{2 \cdot 3} + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.17.2

Multiplica $2$ por $3$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 15 {(3 e^{4 x})}^{2} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.18

Aplica la regla del producto a $3 e^{4 x}$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 15 (3^{2} {(e^{4 x})}^{2}) {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.19

Eleva $3$ a la potencia de $2$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 15 (9 {(e^{4 x})}^{2}) {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.20

Multiplica los exponentes en ${(e^{4 x})}^{2}$ .

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Paso 2.20.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 15 (9 e^{4 x \cdot 2}) {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.20.2

Multiplica $2$ por $4$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 15 (9 e^{8 x}) {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.21

Multiplica $9$ por $15$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 135 e^{8 x} {(\cos^{2} (5 x))}^{4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.22

Multiplica los exponentes en ${(\cos^{2} (5 x))}^{4}$ .

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Paso 2.22.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 135 e^{8 x} {\cos (5 x)}^{2 \cdot 4} + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.22.2

Multiplica $2$ por $4$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 135 e^{8 x} \cos^{8} (5 x) + 6 (3 e^{4 x}) {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.23

Multiplica $3$ por $6$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 135 e^{8 x} \cos^{8} (5 x) + 18 e^{4 x} {(\cos^{2} (5 x))}^{5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.24

Multiplica los exponentes en ${(\cos^{2} (5 x))}^{5}$ .

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Paso 2.24.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 135 e^{8 x} \cos^{8} (5 x) + 18 e^{4 x} {\cos (5 x)}^{2 \cdot 5} + {(\cos^{2} (5 x))}^{6}$

Paso 2.24.2

Multiplica $2$ por $5$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 135 e^{8 x} \cos^{8} (5 x) + 18 e^{4 x} \cos^{10} (5 x) + {(\cos^{2} (5 x))}^{6}$

Paso 2.25

Multiplica los exponentes en ${(\cos^{2} (5 x))}^{6}$ .

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Paso 2.25.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 135 e^{8 x} \cos^{8} (5 x) + 18 e^{4 x} \cos^{10} (5 x) + {\cos (5 x)}^{2 \cdot 6}$

Paso 2.25.2

Multiplica $2$ por $6$ .

$729 e^{24 x} + 1458 e^{20 x} \cos^{2} (5 x) + 1215 e^{16 x} \cos^{4} (5 x) + 540 e^{12 x} \cos^{6} (5 x) + 135 e^{8 x} \cos^{8} (5 x) + 18 e^{4 x} \cos^{10} (5 x) + \cos^{12} (5 x)$