What is the integral of (cos(x))/(sqrt(1+sin^2(x))) ?

The integral of (cos(x))/(sqrt(1+sin^2(x))) is ln|sqrt(sin^2(x)+1)+sin(x)|+C

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integral of (cos(x))/(sqrt(1+sin^2(x)))

\int \frac{cos ( x )}{1 + sin ^{2} ( x )} d x

ln sin^{2} (x) + 1 + sin (x) + C

Solution steps

\int \frac{cos ( x )}{1 + sin ^{2} ( x )} d x

Apply u-substitution

: \int \frac{1}{1 + u ^{2}} d u

= \int \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

Substitute back

u = sin (x)

= ln sin^{2} (x) + 1 + sin (x)

Add a constant to the solution

= ln sin^{2} (x) + 1 + sin (x) + C

x \to 4 - lim (\frac{x + 4}{2 ^{x} - 4})

\int_{3}^{6} (x - 5)^{2} d x

4 y^{^{''}} - y = 0, y (- 2) = 1, y^{^{'}} (- 2) = - 1

y^{^{''}} - y^{^{'}} - 6 y = e^{3 x}

1 - e^{- 5 t}

What is the integral of (cos(x))/(sqrt(1+sin^2(x))) ?
The integral of (cos(x))/(sqrt(1+sin^2(x))) is ln|sqrt(sin^2(x)+1)+sin(x)|+C