What is the integral from 0 to 1 of xtan(x^2) ?

The integral from 0 to 1 of xtan(x^2) is -1/2 ln(cos(1))

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integral from 0 to 1 of xtan(x^2)

\int_{0}^{1} x tan (x^{2}) d x

- \frac{1}{2} ln (cos (1))

Decimal

0.30781 \dots

Solution steps

\int_{0}^{1} x tan (x^{2}) d x

Apply u-substitution

: \int_{0}^{1} \frac{tan ( u )}{2} d u

= \int_{0}^{1} \frac{tan ( u )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{0}^{1} tan (u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot \int_{0}^{1} \frac{sin ( u )}{cos ( u )} d u

Apply u-substitution

: \int_{1}^{c o s (1)} - \frac{1}{v} d v

= \frac{1}{2} \cdot \int_{1}^{c o s (1)} - \frac{1}{v} d v

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= \frac{1}{2} (- \int_{c o s (1)}^{1} - \frac{1}{v} d v)

Take the constant out:

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

= \frac{1}{2} (- (- \int_{c o s (1)}^{1} \frac{1}{v} d v))

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} (- (- [ln ∣ v ∣]_{c o s (1)}^{1}))

Simplify

= \frac{1}{2} [ln ∣ v ∣]_{c o s (1)}^{1}

Compute the boundaries:

- ln (cos (1))

= \frac{1}{2} (- ln (cos (1)))

Simplify

= - \frac{1}{2} ln (cos (1))

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What is the integral from 0 to 1 of xtan(x^2) ?
The integral from 0 to 1 of xtan(x^2) is -1/2 ln(cos(1))