Evaluate the following integral. ∫(5+x+tan2x)dx

Question

Evaluate the following integral.  ∫(5+x+tan2x)dx

Ryan Willis · Accepted Answer

Step 1
We have to evaluate the integral:

\int (5 + x + \tan^{2} x) d x

We will use following formula:

\int (f (x) + g (x)) d x = \int f (x) d x + \int g (x) d x

\int d x = x + c

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

\int \sec^{2} x d x = \tan x + C

\sec^{2} x - 1 = \tan^{2} x

Step 2
Applying above formula, we get

\int (5 + x + \tan^{2} x) d x = \int 5 d x + \int x d x + (\sec^{2} x - 1) d x

= 5 \int d x + \frac{x^{1 + 1}}{1 + 1} + \int \sec^{2} x d x - \int d x

= 5 x + \frac{x^{2}}{2} + \tan x - x + C

= \frac{x^{2}}{2} + 4 x + \tan x + C

Where, C is an arbitrary constant.
Hence, value of given integration is

\frac{x^{2}}{2} + 4 x + \tan x + C

.

Louise Eldridge · Accepted Answer

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=sec2x−1.  5x+x22+∫sec2x−1dx  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+tan⁡x+∫−1dx  Step 7: Use this rule: ∫adx=ax+C.  4x+x22+tan⁡x  Step 8: Add constant.  4x+x22+tan⁡x+C

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

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