Explained: "Evaluate the following derivatives. ddx∫3excos⁡t2dt" - Plainmath

tinfoQ

Answered question

2021-02-13

Evaluate the following derivatives.

\frac{d}{d x} \int_{3}^{e^{x}} {\cos t}^{2} d t

Answer & Explanation

Talisha

Skilled2021-02-15Added 93 answers

Step 1
To evaluate the derivative,

\frac{d}{d x} \int_{3}^{e^{x}} {\cos t}^{2} d t

Step 2
According to the Lebnitz Rule of differentiation under the sign of integration,

\frac{d}{d x} \int_{g (x)}^{h (x)} f (t) d t = f (h (x)) \frac{d h (x)}{d x} - f (g (x)) \frac{d g (x)}{d x}

Step 3
Let us apply the above rule,

\int_{3}^{e^{x}} {\cos t}^{2} d t = {\cos (e^{x})}^{2} \frac{d (e^{x})}{d x} - {\cos (3)}^{2} \frac{d (3)}{d x}

\Rightarrow \int_{3}^{e^{x}} {\cos t}^{2} d t = {\cos e}^{2 x} (e^{x}) - \cos 9 (0)

∴ \int_{3}^{e^{x}} {\cos t}^{2} d t = e^{x} \cos (e^{2 x})

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