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

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

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

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

- ln cos^{2} (x) + 1 + cos (x) + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

= - \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 = cos (x)

= - ln cos^{2} (x) + 1 + cos (x)

Add a constant to the solution

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

\int_{7}^{9} (x^{2} + 2 x - 3) d x

y^{^{''}} + 6 y^{^{'}} + 9 y = 32 e^{t}, y (0) = 1, y^{^{'}} (0) = 7

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

f (x) = ln (1 + e^{x})

∣ x ∣, x^{2} - 6

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