Evaluate the following integral. \int (5+x+\tan^{2}x)dx

jazzcutie0h 2021-11-16 Answered
Evaluate the following integral.
(5+x+tan2x)dx
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Expert Answer

Ryan Willis
Answered 2021-11-17 Author has 15 answers
Step 1
We have to evaluate the integral:
(5+x+tan2x)dx
We will use following formula:
(f(x)+g(x))dx=f(x)dx+g(x)dx
dx=x+c
xndx=xn+1n+1+C
sec2xdx=tanx+C
sec2x1=tan2x
Step 2
Applying above formula, we get
(5+x+tan2x)dx=5dx+xdx+(sec2x1)dx
=5dx+x1+11+1+sec2xdxdx
=5x+x22+tanxx+C
=x22+4x+tanx+C
Where, C is an arbitrary constant.
Hence, value of given integration is x22+4x+tanx+C.
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Louise Eldridge
Answered 2021-11-18 Author has 17 answers
Step 1: Expand.
5+x+tan2xdx
Step 2: Use Sum Rule: f(x)+g(x)dx=f(x)dx+g(x)dx.
5+xdx+tan2xdx
Step 3: Use Power Rule: xndx=xn+1n+1+C.
5x+x22+tan2xdx
Step 4: Use Pythagorean Identities: tan2x=sec2x1.
5x+x22+sec2x1dx
Step 5: Use Sum Rule: f(x)+g(x)dx=f(x)dx+g(x)dx.
5x+x22+sec2xdx+1dx
Step 6: The derivative of \tan x is sec2x.
5x+x22+tanx+1dx
Step 7: Use this rule: adx=ax+C.
4x+x22+tanx
Step 8: Add constant.
4x+x22+tanx+C
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