What is the integral from 0 to pi/4 of 7xsec^2(x) ?

The integral from 0 to pi/4 of 7xsec^2(x) is 7(pi/4-1/2 ln(2))

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integral from 0 to pi/4 of 7xsec^2(x)

\int_{0}^{\frac{π}{4}} 7 x sec^{2} (x) d x

7 (\frac{π}{4} - \frac{1}{2} ln (2))

Decimal Notation

3.07177 \dots

Solution steps

\int_{0}^{\frac{π}{4}} 7 x sec^{2} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 7 \cdot \int_{0}^{\frac{π}{4}} x sec^{2} (x) d x

Apply Integration By Parts

: [x tan (x) - \int tan (x) d x]_{0}^{\frac{π}{4}}

= 7 [x tan (x) - \int tan (x) d x]_{0}^{\frac{π}{4}}

\int tan (x) d x = - ln ∣ cos (x) ∣

= 7 [x tan (x) - (- ln ∣ cos (x) ∣)]_{0}^{\frac{π}{4}}

Simplify

= 7 [x tan (x) + ln ∣ cos (x) ∣]_{0}^{\frac{π}{4}}

Compute the boundaries:

\frac{π}{4} - \frac{1}{2} ln (2)

= 7 (\frac{π}{4} - \frac{1}{2} ln (2))

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What is the integral from 0 to pi/4 of 7xsec^2(x) ?
The integral from 0 to pi/4 of 7xsec^2(x) is 7(pi/4-1/2 ln(2))