Trigonometric Integrals

Even Powers of sin x or cos x:

Apply the sine and cosine half-angle:

Trigonometric Integrals

Cosine Integral

Integration

Integration

Integration

Integration

Integration

Integration

Integration Solution

Odd Powers of sin x or cos x:

Relate sine and cosine using the formula:

Relate Sine and Cosine

Integral of Sine to x Cubed

Integration

Integration Solution

Integral of Cosine to x Cubed

Integration

Integration

Integration Solution

Integral of Cosine to x 5

Integration

Integration

Integration Solution


With an Even and Odd Exponent:

The odd exponent becomes one even and one odd.

Trigonometric Integral

Integration

Integration

Integration

Integration

Integration Solution


The change of variable can also be made t = sin x or t = cos x

Trigonometric Integral

Integration

Integration

Integration Solution

Trigonometric Integral

Change of Variable

Integration

Integration

Integration

Integration Solution

Trigonometric Integral

Change of Variable

Integration

Integration Solution


Products of Type sin(nx) · cos(mx):

Products are transformed into sums:

Transformations from Products to Sums

Transformations from Products to Sums

Transformations from Products to Sums

Transformations from Products to Sums


Trigonometric Integral

Integral Solution

Trigonometric Integral

Integral

Integral Solution

Trigonometric Integral

Integral

Integral Solution

Trigonometric Integral

Integral

cos (-4x) = cos 4x

Integral Solution