What is the integral from 0 to pi/2 of (cos(t))/(sqrt(1+sin^2(t))) ?

The integral from 0 to pi/2 of (cos(t))/(sqrt(1+sin^2(t))) is ln(1+sqrt(2))

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integral from 0 to pi/2 of (cos(t))/(sqrt(1+sin^2(t)))

\int_{0}^{\frac{π}{2}} \frac{cos ( t )}{1 + sin ^{2} ( t )} d t

ln (1 + 2)

Decimal

0.88137 \dots

Solution steps

\int_{0}^{\frac{π}{2}} \frac{cos ( t )}{1 + sin ^{2} ( t )} d t

Apply u-substitution

= \int_{0}^{1} \frac{1}{1 + u ^{2}} d u

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = ln u^{2} + 1 + u

= [ln u^{2} + 1 + u]_{0}^{1}

Compute the boundaries:

ln (1 + 2)

= ln (1 + 2)

What is the integral from 0 to pi/2 of (cos(t))/(sqrt(1+sin^2(t))) ?
The integral from 0 to pi/2 of (cos(t))/(sqrt(1+sin^2(t))) is ln(1+sqrt(2))