Question

Integrate $3^x.$

Answer 1

To solve the integral use the substitution method

Given: $3^x$

Let $u=3^x$

Taking log on both sides we get,

$\ln u=\ln \left(3^x\right)$

$\ln u=x\ln \left(3\right)$

Now, $u=e^{xln3}$

So,

$\int 3^{x}dx=\int e^{x\ln 3}dx=\frac{e^{x\ln 3}}{\ln 3}+c$

or, $=\frac{e^{\ln 3^x}}{\ln 3}+c=\frac{3^x}{\ln 3}+c\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{(}\mathbf{\because }{\mathit{e}}^{\mathbf{ln}\mathbf{}\mathbf{a}}\mathbf{=}\mathit{a}\mathbf{)}$

Hence, integral of $3^x$ is $\frac{e^{x\ln 3}}{\ln 3}+c$.. or $\frac{3^x}{\ln 3}+c$