What is the integral of (5sin(x))/(cos^2(x)) ?

The integral of (5sin(x))/(cos^2(x)) is 5sec(x)+C

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integral of (5sin(x))/(cos^2(x))

\int \frac{5 sin ( x )}{cos ^{2} ( x )} d x

5 sec (x) + C

Solution steps

\int \frac{5 sin ( x )}{cos ^{2} ( x )} d x

Take the constant out:

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

= 5 \cdot \int \frac{sin ( x )}{cos ^{2} ( x )} d x

Apply u-substitution

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

= 5 \cdot \int - \frac{1}{u ^{2}} d u

Take the constant out:

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

= 5 (- \int \frac{1}{u ^{2}} d u)

Apply the Power Rule

: - \frac{1}{u}

= 5 (- (- \frac{1}{u}))

Substitute back

u = cos (x)

= 5 (- (- \frac{1}{cos ( x )}))

Simplify

5 (- (- \frac{1}{cos ( x )})) : 5 sec (x)

= 5 sec (x)

Add a constant to the solution

= 5 sec (x) + C

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What is the integral of (5sin(x))/(cos^2(x)) ?
The integral of (5sin(x))/(cos^2(x)) is 5sec(x)+C