Integral of cos⁶(x): Calculator and Formula

The integral of cos⁶(x) calculator evaluates the indefinite integral of cos⁶(x) with respect to x and returns an antiderivative plus its derivative check. Enter an angle in radians; the standard identity gives a formula that works for any real x, with an arbitrary constant of integration.

Formula

Using radians, an antiderivative of cos⁶(x) is:

∫ cos⁶(x) dx = 5x/16 + 15 sin(2x)/64 + 3 sin(4x)/128 + sin(6x)/192 + C.

• x is the angle, measured in radians.
• sin is the sine function, with its argument in radians.
• C is an arbitrary constant, because differentiating any constant gives zero.

The calculator displays the antiderivative without C, since a calculator cannot select one particular constant for an indefinite integral. Its second output is the integrand recovered by differentiating the displayed expression.

Worked Example

For the default input x = 1 radian, substitute 1 into the antiderivative:

1. 5(1)/16 = 0.3125.
2. 15 sin(2)/64 ≈ 0.34102.
3. 3 sin(4)/128 ≈ −0.01772.
4. sin(6)/192 ≈ −0.00145.
5. Add the terms: the result is approximately 0.63435.

The exact calculator expression at this input is 5/16 + 15 sin(2)/64 + 3 sin(4)/128 + sin(6)/192. The derivative check is cos⁶(1) ≈ 0.1135. The decimal antiderivative is rounded to six places; the check is rounded separately to six places.

How the Power-Reduction Method Works

A direct substitution is not especially useful here: if u = cos(x), then du = −sin(x) dx, and the original expression does not contain a factor of sin(x) to complete that substitution. A reliable approach is to rewrite an even power of cosine using a multiple-angle identity.

Start with cos²(x) = (1 + cos(2x))/2. Squaring that identity gives cos⁴(x) = (3 + 4 cos(2x) + cos(4x))/8. Multiplying by cos²(x), then applying product-to-sum identities to the products of cosines, produces:

cos⁶(x) = (10 + 15 cos(2x) + 6 cos(4x) + cos(6x))/32.

Now integrate each term. The constant term integrates to a term proportional to x, while ∫cos(kx) dx = sin(kx)/k for nonzero k. Thus the coefficients become:

• 10/32 becomes 5x/16.
• (15/32)cos(2x) becomes 15 sin(2x)/64.
• (6/32)cos(4x) becomes 3 sin(4x)/128.
• (1/32)cos(6x) becomes sin(6x)/192.

This identity is useful beyond integration: it shows that the sixth power has a constant average component plus oscillations at twice, four times, and six times the original angle.

Using the Calculator and Reading the Result

Enter x in radians. The default is 1 radian, and the input permits any real number, including negative angles. The first output is an antiderivative evaluated at your input; the second is the value of cos⁶(x), which is the derivative of the antiderivative.

An indefinite integral is technically a family of functions, not one number. The calculator shows the particular antiderivative obtained by taking C = 0. If the task is to find a definite integral from a to b, evaluate the antiderivative at both endpoints and subtract:

∫ₐᵇ cos⁶(x) dx = F(b) − F(a),

where F is the expression in the formula. The constants cancel. Both endpoints must use the same angle unit, and this formula assumes radians.

For example, the integral over one full cycle, from 0 to 2π, is 5π/8. The sine terms vanish at both endpoints, leaving the linear term. Dividing by the cycle length 2π gives an average value of 5/16 for cos⁶(x) over a complete cycle.

Common Mistakes and Limits

• Mixing degrees and radians: Calculus identities and the displayed formula use radians. If a problem gives degrees, convert first: x radians = x degrees × π/180. A degree-mode calculator can otherwise produce a misleading comparison.
• Forgetting the chain rule: The derivative of sin(2x) is 2cos(2x), not just cos(2x). Each frequency in the identity requires division by that frequency when integrated.
• Leaving out C: Every indefinite integral has an arbitrary constant. The calculator omits it in the displayed numeric expression, but it belongs in a complete symbolic answer.
• Confusing an antiderivative value with a definite integral: A single value F(x) is not the accumulated integral over an interval. Subtract the endpoint values for a definite integral.
• Rounding too early: Keep exact fractions and symbolic sine terms until the final decimal evaluation, especially when subtracting close endpoint values.

The calculator evaluates a standard identity; it does not resolve a user-specific integration constant, choose bounds, or infer whether an angle was intended in degrees. For definite integrals, numerical tools may also be useful when an integrand is more complicated, but here the exact antiderivative is available.

Why the Average Over a Cycle Is 5/16

The average value over an interval of length 2π is the integral over that interval divided by 2π. Since the oscillating sine terms contribute zero net change over a full cycle, only 5x/16 contributes. Across 0 to 2π, that term changes by 5(2π)/16 = 5π/8. Dividing by 2π gives 5/16.

This is also a useful check on the power-reduction identity: the constant term in cos⁶(x) is 10/32, which simplifies to 5/16. The average of each nonconstant cosine term over a full cycle is zero. Do not confuse this average with cos⁶ evaluated at the average angle; averaging a function and evaluating it at an average input are generally different operations.

Frequently Asked Questions

What is the integral of cos⁶(x)?
The indefinite integral is 5x/16 + 15 sin(2x)/64 + 3 sin(4x)/128 + sin(6x)/192 + C, with x in radians.

Why does the answer include a term with x?
Power reduction creates a constant term, 5/16, in cos⁶(x). Integrating that constant produces 5x/16.

Does the calculator include the constant of integration?
No. It displays one antiderivative with C set to zero. Add + C when writing the general indefinite integral.

How do I calculate the definite integral from a to b?
Substitute b and a into the antiderivative and compute F(b) − F(a). Use radians for both endpoints.

What is the average value of cos⁶(x) over a full period?
It is 5/16 over any interval spanning one complete cycle, such as 0 to 2π.

Can I enter x in degrees?
Not directly with this formula. Convert degrees to radians by multiplying by π/180 before evaluating the sine terms.

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in x | Angle x | rad | 1 | -
out integral | Integral of cos^6(x) | rad | 5*x/16 + 15*sin(2*x)/64 + 3*sin(4*x)/128 + sin(6*x)/192 | 6
out derivative | Derivative check | - | cos(x)^6 | 6