pancha3

2021-10-13

$\int \left({x}^{a}+{a}^{x}+{a}^{a}\right)dx=$
a.$\frac{{x}^{a+1}}{a+1}+\frac{{a}^{x}}{\mathrm{ln}a}+c$
b.$\frac{{x}^{a+1}}{a+1}+\frac{{a}^{x}}{\mathrm{ln}a}+{a}^{a}x+c$
c.$\frac{{x}^{a+1}}{a+1}{a}^{x}\mathrm{ln}a+{a}^{a}x+c$
d.$\frac{{x}^{a+1}}{a+1}+\frac{{a}^{x}}{\mathrm{ln}a}+{a}^{a}x$

Brittany Patton

Step 1
Consider the given integral.
$\int \left({x}^{a}+{a}^{x}+{a}^{a}\right)dx$
Rewrite the given integral as separate integrals.
$\int \left({x}^{a}+{a}^{x}+{a}^{a}\right)dx=\int {x}^{a}dx+\int {a}^{x}dx+\int {a}^{a}dx$
Step 2
Integrate using the formulas:
$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+c$
$\int {a}^{x}dx=\frac{{a}^{x}}{\mathrm{ln}a}+c$
$\int {a}^{n}dx={a}^{n}x+c$
Using these formulae, the given integral can be integrated as,
$\int \left({x}^{a}+{a}^{x}+{a}^{a}\right)dx=\frac{{x}^{a+1}}{a+1}+\frac{{a}^{x}}{\mathrm{ln}a}+{a}^{a}x+C$
Therefore, the correct option is B.

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