Find the indefinite integral. (use c for the constant of integration.) sech8 x tanh x dx

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Space Space · Answer 1 · 2021-07-26T03:24:08+02:00

Answer:

[tex]\displaystyle \int {sech^8(x)tanh(x)} \, dx = -\frac{sech^8(x)}{8} + C[/tex]

General Formulas and Concepts:

Calculus

Differentiation

Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]

Integration

Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle \int {sech^8(x)tanh(x)} \, dx[/tex]

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

Set u: [tex]\displaystyle u = sech^8(x)[/tex]
[u] Differentiate [Hyperbolic Differentiation, Chain Rule]: [tex]\displaystyle du = -8sech^8(x)tanh(x) \ dx[/tex]

Step 3: integrate Pt. 2

[Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {sech^8(x)tanh(x)} \, dx = \frac{-1}{8}\int {-8sech^8(x)tanh(x)} \, dx[/tex]
[Integral] U-Substitution: [tex]\displaystyle \int {sech^8(x)tanh(x)} \, dx = \frac{-1}{8}\int {} \, du[/tex]
[Integral] Reverse Power Rule: [tex]\displaystyle \int {sech^8(x)tanh(x)} \, dx = \frac{-u}{8} + C[/tex]
Back-Substitute: [tex]\displaystyle \int {sech^8(x)tanh(x)} \, dx = -\frac{sech^8(x)}{8} + C[/tex]

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration