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Integral of 1cos⁡(x)2\frac{1}{\cos\left(x\right)^{2}}

The integral of 1cos⁡(x)2\frac{1}{\cos\left(x\right)^{2}} is tan⁡(x)+C\tan\left(x\right) + C.

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Graph of f(x) = 1 / cos(x)^2 with the area from 0 to 1 shaded-0.4-0.20.20.40.60.811.21.41.62004006008001000f(x)
Graph of f(x) = 1 / cos(x)^2 with the area from 0 to 1 shaded. The shaded area equals the definite integral, about 1.5574077.

How to work it out

  1. 1

    We want an antiderivative of f(x) = 1 / cos(x)^2.

    ∫1cos⁡(x)2 dx\int \frac{1}{\cos\left(x\right)^{2}}\,dx
  2. 2

    Method: standard integral.

  3. 3

    That gives the antiderivative; add the constant C because any constant disappears when you differentiate:

    ∫1cos⁡(x)2 dx=tan⁡(x)+C\int \frac{1}{\cos\left(x\right)^{2}}\,dx = \tan\left(x\right) + C
  4. 4

    Check by differentiating the answer; you must get the original function back:

    ddx[tan⁡(x)]=1cos⁡(x)2\frac{d}{dx}\left[\tan\left(x\right)\right] = \frac{1}{\cos\left(x\right)^{2}}
  5. 5

    As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and 1:

    ∫011cos⁡(x)2 dx=[tan⁡(x)]01≈1.5574077\int_{0}^{1} \frac{1}{\cos\left(x\right)^{2}}\,dx = \Big[\tan\left(x\right)\Big]_{0}^{1} \approx 1.5574077
LaTeX source of the formulas on this page
\[ \int \frac{1}{\cos\left(x\right)^{2}}\,dx = \tan\left(x\right) + C \]
\[ \int \frac{1}{\cos\left(x\right)^{2}}\,dx \]
\[ \frac{d}{dx}\left[\tan\left(x\right)\right] = \frac{1}{\cos\left(x\right)^{2}} \]
\[ \int_{0}^{1} \frac{1}{\cos\left(x\right)^{2}}\,dx = \Big[\tan\left(x\right)\Big]_{0}^{1} \approx 1.5574077 \]

Definite integrals of the same function

IntervalValue
from 0 to 11.5574077
from 0 to 0.50.54630249

Questions

What is the integral of 1 / cos(x)^2?

The integral of 1cos⁡(x)2\frac{1}{\cos\left(x\right)^{2}} is tan⁡(x)+C\tan\left(x\right) + C.

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of 1 / cos(x)^2 from 0 to 1?

About 1.5574077.

More integrals

    Integral of 1 / cos(x)^2: tan(x) + C (with steps) | WordGPT