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

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

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

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

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

Solution steps

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

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 (t)

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{7}{( x + 1 ) ^{2}})

\frac{d}{d x} (\frac{1}{1 + \frac{1}{4} x})

\int (a x^{4} + b x^{3} + 3 c) d x

\frac{\partial}{\partial x} (\frac{( 2 π \cdot l ^{2} )}{x})

\int_{0}^{16} x d x

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