What is the integral of (2sin(x))/(cos^3(x)) ?

The integral of (2sin(x))/(cos^3(x)) is tan^2(x)+C

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integral of (2sin(x))/(cos^3(x))

\int \frac{2 sin ( x )}{cos ^{3} ( x )} d x

tan^{2} (x) + C

Solution steps

\int \frac{2 sin ( x )}{cos ^{3} ( x )} d x

Take the constant out:

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

= 2 \cdot \int \frac{sin ( x )}{cos ^{3} ( x )} d x

Rewrite using trig identities

= 2 \cdot \int tan (x) sec^{2} (x) d x

Apply u-substitution

: \int u d u

= 2 \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= 2 \cdot \frac{u ^{2}}{2}

Substitute back

u = tan (x)

= 2 \cdot \frac{tan ^{2} ( x )}{2}

Simplify

2 \cdot \frac{tan ^{2} ( x )}{2} : tan^{2} (x)

= tan^{2} (x)

Add a constant to the solution

= tan^{2} (x) + C

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What is the integral of (2sin(x))/(cos^3(x)) ?
The integral of (2sin(x))/(cos^3(x)) is tan^2(x)+C