Question: "Evaluate the integral. ∫(ex−e−x)(ex+e−x)3dx"

Lucille Smitherman

Answered question

2021-11-18

Evaluate the integral.

\int (e^{x} - e^{- x}) {(e^{x} + e^{- x})}^{3} d x

Supoilign1964

Beginner2021-11-19Added 19 answers

Let substitute

t = e^{x} + e^{- x}

.
Differentiating both sides, we get

\frac{d t}{d x} = \frac{d}{d x} (e^{x} + e^{- x})

\frac{d t}{d x} = e^{x} - e^{- x}

d t = (e^{x} - e^{- x}) d x

Step 3
Substituting the value in the given integral.

\int (e^{x} - e^{- x}) {(e^{x} + e^{- x})}^{3} d x = \int t^{3} d t

= \frac{t^{4}}{4} + C

= \frac{{(e^{x} + e^{- x})}^{4}}{4} + C

[

t = e^{x} + e^{- x}

]
Hence,

\int (e^{x} - e^{- x}) {(e^{x} + e^{- x})}^{3} d x = \frac{{(e^{x} + e^{- x})}^{4}}{4} + C

James Obrien

Beginner2021-11-20Added 16 answers

Step 1: Remove parentheses.

\int (e^{x} - e^{- x}) {(e^{x} + e^{- x})}^{3} d x

Step 2: Use Integration by Substitution.
Let

u = e^{x} + e^{- x}, d u = e^{x} - e^{- x} d x

Step 3: Using u and du above, rewrite

\int (e^{x} - e^{- x}) {(e^{x} + e^{- x})}^{3} d x

\int u^{3} d u

Step 4: Use Power Rule:

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

\frac{u^{4}}{4}

Step 5: Substitute

u = e^{x} + e^{- x}

back into the original integral.

\frac{{(e^{x} + e^{- x})}^{4}}{4}

Step 6: Add constant.

\frac{{(e^{x} + e^{- x})}^{4}}{4} + C

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