 # Evaluate the following integral. \int (5+x+\tan^{2}x)dx jazzcutie0h 2021-11-16 Answered
Evaluate the following integral.
$\int \left(5+x+{\mathrm{tan}}^{2}x\right)dx$
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Step 1
We have to evaluate the integral:
$\int \left(5+x+{\mathrm{tan}}^{2}x\right)dx$
We will use following formula:
$\int \left(f\left(x\right)+g\left(x\right)\right)dx=\int f\left(x\right)dx+\int g\left(x\right)dx$
$\int dx=x+c$
$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$
$\int {\mathrm{sec}}^{2}xdx=\mathrm{tan}x+C$
${\mathrm{sec}}^{2}x-1={\mathrm{tan}}^{2}x$
Step 2
Applying above formula, we get
$\int \left(5+x+{\mathrm{tan}}^{2}x\right)dx=\int 5dx+\int xdx+\left({\mathrm{sec}}^{2}x-1\right)dx$
$=5\int dx+\frac{{x}^{1+1}}{1+1}+\int {\mathrm{sec}}^{2}xdx-\int dx$
$=5x+\frac{{x}^{2}}{2}+\mathrm{tan}x-x+C$
$=\frac{{x}^{2}}{2}+4x+\mathrm{tan}x+C$
Where, C is an arbitrary constant.
Hence, value of given integration is $\frac{{x}^{2}}{2}+4x+\mathrm{tan}x+C$.
###### Not exactly what you’re looking for? Louise Eldridge
Step 1: Expand.
$\int 5+x+{\mathrm{tan}}^{2}xdx$
Step 2: Use Sum Rule: $\int f\left(x\right)+g\left(x\right)dx=\int f\left(x\right)dx+\int g\left(x\right)dx$.
$\int 5+xdx+\int {\mathrm{tan}}^{2}xdx$
Step 3: Use Power Rule: $\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$.
$5x+\frac{{x}^{2}}{2}+\int {\mathrm{tan}}^{2}xdx$
Step 4: Use Pythagorean Identities: ${\mathrm{tan}}^{2}x={\mathrm{sec}}^{2}x-1$.
$5x+\frac{{x}^{2}}{2}+\int {\mathrm{sec}}^{2}x-1dx$
Step 5: Use Sum Rule: $\int f\left(x\right)+g\left(x\right)dx=\int f\left(x\right)dx+\int g\left(x\right)dx$.
$5x+\frac{{x}^{2}}{2}+\int {\mathrm{sec}}^{2}xdx+\int -1dx$
Step 6: The derivative of \tan x is ${\mathrm{sec}}^{2}x$.
$5x+\frac{{x}^{2}}{2}+\mathrm{tan}x+\int -1dx$
Step 7: Use this rule: $\int adx=ax+C$.
$4x+\frac{{x}^{2}}{2}+\mathrm{tan}x$
$4x+\frac{{x}^{2}}{2}+\mathrm{tan}x+C$