integral of (5x)/(cos^2(x^2)) — step-by-step solution

Let's compute the integral:

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

For the math expression:

\frac{5 x}{{\cos (x^{2})}^{2}}

, since (tan(a))' = a'/cos(a)^2, we try differentiating

\tan (x^{2})

Let's differentiate:

\frac{d}{d x} (\tan (x^{2}))

We differentiate using:

\frac{d}{d x} (\tan (f)) = \frac{d}{d x} (f) \cdot \frac{1}{{\cos (f)}^{2}}

(where f =

x^{2}

)

ⓘ The derivative of tan(x) is 1/cos(x)^2:

\frac{d}{d x} (\tan (x^{2}))

\frac{d}{d x} (x^{2}) \cdot \frac{1}{{\cos (x^{2})}^{2}}

\frac{d}{d x} (x^{2}) \cdot \frac{1}{{\cos (x^{2})}^{2}}

ⓘ The derivative of x^n is n*x^(n-1):

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

2 x

\frac{2 x}{{\cos (x^{2})}^{2}}

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

ⓘ The derivative of

\frac{5}{2} \cdot \tan (x^{2})

\frac{5 x}{{\cos (x^{2})}^{2}}

, so

\frac{5}{2} \cdot \tan (x^{2})

is an antiderivative of

\frac{5 x}{{\cos (x^{2})}^{2}}

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

\frac{5}{2} \cdot \tan (x^{2})

\frac{5}{2} \cdot \tan (x^{2})

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{5}{2} \cdot \tan (x^{2})

\frac{5 \cdot \tan (x^{2})}{2}

\frac{5 \cdot \tan (x^{2})}{2}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

\frac{5 \cdot \tan (x^{2})}{2} + C