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Calculus and Analysis
Integral Calculus
Integrals

e1s2kat26

Answered question

2021-10-09

Use the trigonometric rules for integrals to solve the following integrals. for b and c
b)

\int_{0}^{1} 2 π x \cdot \sin (π x^{2}) d x

\int \frac{\sec^{2} (\sqrt{x})}{\sqrt{x}} d x

Answer & Explanation

firmablogF

Skilled2021-10-10Added 92 answers

b) $\int_{0}^{1} 2 π x \sin (π x^{2}) d x$
put $π x^{2} = t$
$π 2 x d x = d t$
$x = 0 t = 0$
$x = 1 t = π$
$= \int_{0}^{π} \sin t d t$
$= - {[\cos t]}_{0}^{π}$
$= - [\cos π - \cos 0]$
$= - [(- 1) - 1]$
$= - (- 2) = 2$
c) $\int \frac{\sec^{2} \sqrt{x}}{\sqrt{x}} d x$
put $\sqrt{x} = t$
$\frac{1}{2 \sqrt{x}} d x = d t$
$= \int \sec^{2} t 2 \cdot d t$
$= 2 \int \sec^{2} t d t$
$= 2 \tan t + C$
$= 2 \tan (\sqrt{x}) + C$

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