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

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

Supoilign1964 · Accepted Answer

Let substitute t=ex+e−x.  Differentiating both sides, we get  dtdx=ddx(ex+e−x)  dtdx=ex−e−x  dt=(ex−e−x)dx  Step 3  Substituting the value in the given integral.  ∫(ex−e−x)(ex+e−x)3dx=∫t3dt  =t44+C  =(ex+e−x)44+C   [t=ex+e−x]  Hence, ∫(ex−e−x)(ex+e−x)3dx=(ex+e−x)44+C

James Obrien · Accepted Answer

Step 1: Remove parentheses.  ∫(ex−e−x)(ex+e−x)3dx  Step 2: Use Integration by Substitution.  Let u=ex+e−x,du=ex−e−xdx  Step 3: Using u and du above, rewrite ∫(ex−e−x)(ex+e−x)3dx.  ∫u3du  Step 4: Use Power Rule: ∫xndx=xn+1n+1+C.  u44  Step 5: Substitute u=ex+e−x back into the original integral.  (ex+e−x)44  Step 6: Add constant.  (ex+e−x)44+C

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

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