What is the integral of (cos(x))/(7+sin^2(x)) ?

The integral of (cos(x))/(7+sin^2(x)) is 1/(sqrt(7))arctan((sin(x))/(sqrt(7)))+C

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integral of (cos(x))/(7+sin^2(x))

\int \frac{cos ( x )}{7 + sin ^{2} ( x )} d x

\frac{1}{7} arctan (\frac{sin ( x )}{7}) + C

Solution steps

\int \frac{cos ( x )}{7 + sin ^{2} ( x )} d x

Apply u-substitution

: \int \frac{1}{7 + u ^{2}} d u

= \int \frac{1}{7 + u ^{2}} d u

Apply Integral Substitution

= \int \frac{1}{7 ( 1 + v ^{2} )} d v

Take the constant out:

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

= \frac{1}{7} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

\int \frac{1}{1 + v ^{2}} d v = arctan (v)

= \frac{1}{7} arctan (v)

Substitute back

= \frac{1}{7} arctan (\frac{sin ( x )}{7})

Add a constant to the solution

= \frac{1}{7} arctan (\frac{sin ( x )}{7}) + C

f (x) = (x + 2)^{\frac{2}{3}} (2 x^{2} - 5)^{3}

2 - ∣ x ∣, x^{2}

\frac{d y}{d x} + \frac{2 y}{x} = \frac{9}{x}

\frac{\partial}{\partial x} (x^{2} - x y + y^{2})

\frac{d}{d x} (sin (e^{x^{2}}))

What is the integral of (cos(x))/(7+sin^2(x)) ?
The integral of (cos(x))/(7+sin^2(x)) is 1/(sqrt(7))arctan((sin(x))/(sqrt(7)))+C