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g(x)=int_(0)^(x)sqrt(5+4tan t)dt

g^(')((pi)/(4))=

$g(x)=\int_{0}^{x} \sqrt{5+4 \tan t} d t$ $\newline$ $g^{\prime}\left(\frac{\pi}{4}\right)=$

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Evaluate definite integrals using the power rule

Full solution

Q.

g(x)=\int_{0}^{x} \sqrt{5+4 \tan t} d t

\newline

g^{\prime}\left(\frac{\pi}{4}\right)=

Apply Fundamental Theorem of Calculus: Use the Fundamental Theorem of Calculus which states that if $g(x) = \int_{a}^{x} f(t) \, dt$ , then $g'(x) = f(x)$ .
Evaluate integrand at $x = \frac{\pi}{4}$ : Evaluate the integrand at $x = \frac{\pi}{4}$ : $f\left(\frac{\pi}{4}\right) = \sqrt{5 + 4\tan\left(\frac{\pi}{4}\right)}$ .
Substitute into integrand: Since $\tan(\frac{\pi}{4}) = 1$ , substitute into the integrand: $\sqrt{5 + 4(1)} = \sqrt{5 + 4}$ .
Simplify the expression: Simplify the expression: $\sqrt{9} = 3$ .
Final result: Therefore, $g'(\frac{\pi}{4}) = 3$ .

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