What is the integral of-sin(x)tan^2(x) ?

The integral of-sin(x)tan^2(x) is -(sec(x)+cos(x))+C

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integral of-sin(x)tan^2(x)

\int - sin (x) tan^{2} (x) d x

- (sec (x) + cos (x)) + C

Solution steps

\int - sin (x) tan^{2} (x) d x

Take the constant out:

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

= - \int sin (x) tan^{2} (x) d x

Rewrite using trig identities

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

Apply u-substitution

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int 1 d u = u

= - (- (- \frac{1}{u}) + u)

Substitute back

u = cos (x)

= - (- (- \frac{1}{cos ( x )}) + cos (x))

Simplify

- (- (- \frac{1}{cos ( x )}) + cos (x)) : - (sec (x) + cos (x))

= - (sec (x) + cos (x))

Add a constant to the solution

= - (sec (x) + cos (x)) + C

What is the integral of-sin(x)tan^2(x) ?
The integral of-sin(x)tan^2(x) is -(sec(x)+cos(x))+C