Evaluate the following integrals. intfrac{e^{sin x}}{sec x}dx

Question
Integrals
asked 2021-02-14
Evaluate the following integrals.
\(\int\frac{e^{\sin x}}{\sec x}dx\)

Answers (1)

2021-02-15
\(\text{The given integral is,}\)
\(I=\int\frac{e^{\sin x}}{\sec x}dx\)
\(\text{Using trigonometric identity, }\cos\theta=\frac{1}{\sec\theta}\)
\(I=\int\cos x(e^{\sin x})dx\)
\(\text{Using substitution method,}\)
\(e^{\sin x}=t\)
\(\frac{d}{dx}(e^{\sin x})=\frac{dt}{dx}\)
Let\([\frac{d}{d(\sin x)}(e^{\sin x})][\frac{d}{dx}(\sin x)]=\frac{dt}{dx}\)
\((e^{\sin x})(\cos x)=\frac{dt}{dx}\)
\((e^{\sin x})(\cos x)dx=dt\)
\(\text{Then the given integral gets transformed as, }\)
\(I=\int dt=t+C\)
\(\text{Putting back the value of t, we get}\)
\(I=e^{\sin x}+C\)
\(\text{Therefore, the value of the given integral is }e^{\sin x}+C\)
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