\(\text{It is given that}\)

\(\int_{-1}^1 10^xdx\)

\(\text{The given integration can be written as}\)

\(I=\int_{-1}^1 e^{x\ln10}dx\)

\(t=x\ln10\)

\(dx=\frac{1}{\ln10}dt\)

\(\text{The integration wiil be}\)

\(I=\frac{1}{\ln10}\int_{-1}^1 e^tdt\)

\(I=\frac{1}{\ln10}[e^t]_{-1}^1\)

\(\text{Put }t=x\ln10\)

\(I=\frac{1}{\ln10}[10^x]_{-1}^1\)

\(I=\frac{1}{\ln10}[10^1-10^{-1}]\)

\(=\frac{99}{10\ln10}\)

\(\text{Hence the value of integration is }\frac{99}{10\ln10}\)

\(\int_{-1}^1 10^xdx\)

\(\text{The given integration can be written as}\)

\(I=\int_{-1}^1 e^{x\ln10}dx\)

\(t=x\ln10\)

\(dx=\frac{1}{\ln10}dt\)

\(\text{The integration wiil be}\)

\(I=\frac{1}{\ln10}\int_{-1}^1 e^tdt\)

\(I=\frac{1}{\ln10}[e^t]_{-1}^1\)

\(\text{Put }t=x\ln10\)

\(I=\frac{1}{\ln10}[10^x]_{-1}^1\)

\(I=\frac{1}{\ln10}[10^1-10^{-1}]\)

\(=\frac{99}{10\ln10}\)

\(\text{Hence the value of integration is }\frac{99}{10\ln10}\)