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Calculus · Definite-Integrals

Integrate sin(t)^2·cos(t)^2 from 0 to 2·pi

\int_{0}^{2 \pi} \sin^{2}{\left(t \right)} \cos^{2}{\left(t \right)}\, dt

Step by step

\int_{0}^{2 \pi} \sin^{2}{\left(t \right)} \cos^{2}{\left(t \right)}\, dt
First find an antiderivative F, then evaluate F(b) − F(a).
\sin^{2}{\left(t \right)} \cos^{2}{\left(t \right)} = \left(\frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}\right) \left(\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}\right)
Rewrite the integrand into a friendlier form.
u = 2 t,\quad du = 2\, dt
Substitute u = 2 t.
\int \left(\frac{1}{2} - \frac{\cos{\left(2 t \right)}}{2}\right) \left(\frac{\cos{\left(2 t \right)}}{2} + \frac{1}{2}\right)\, dt = \int \frac{1}{8} - \frac{\cos^{2}{\left(u \right)}}{8}\, d_u
Rewrite the integral in terms of u.
\int \frac{1}{8} - \frac{\cos^{2}{\left(u \right)}}{8}\, d_u = \int \frac{1}{8}\, d_u + \int - \frac{\cos^{2}{\left(u \right)}}{8}\, d_u
The integral of a sum is the sum of the integrals.
\int \frac{1}{8}\, d_u = \frac{u}{8}
The integral of a constant c is c·x.
\int - \frac{\cos^{2}{\left(u \right)}}{8}\, d_u = - \frac{1}{8} \int \cos^{2}{\left(u \right)}\, d_u
Pull the constant - \frac{1}{8} out of the integral.
\cos^{2}{\left(u \right)} = \frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}
Rewrite the integrand into a friendlier form.
\int \frac{\cos{\left(2 u \right)}}{2} + \frac{1}{2}\, d_u = \int \frac{\cos{\left(2 u \right)}}{2}\, d_u + \int \frac{1}{2}\, d_u
The integral of a sum is the sum of the integrals.
\int \frac{\cos{\left(2 u \right)}}{2}\, d_u = \frac{1}{2} \int \cos{\left(2 u \right)}\, d_u
Pull the constant \frac{1}{2} out of the integral.
u = 2 u,\quad du = 2\, d_u
Substitute u = 2 u.
\int \cos{\left(2 u \right)}\, d_u = \int \frac{\cos{\left(u \right)}}{2}\, d_u
Rewrite the integral in terms of u.
\int \frac{\cos{\left(u \right)}}{2}\, d_u = \frac{1}{2} \int \cos{\left(u \right)}\, d_u
Pull the constant \frac{1}{2} out of the integral.
\int \cos{\left(u \right)}\, d_u = \sin{\left(u \right)}
Standard trigonometric antiderivative.
= \frac{\sin{\left(2 u \right)}}{2}
Substitute back u = 2 u.
\int \frac{1}{2}\, d_u = \frac{u}{2}
The integral of a constant c is c·x.
= \frac{t}{8} - \frac{\sin{\left(4 t \right)}}{32}
Substitute back u = 2 t.
F(2 \pi) - F(0) = \left(\frac{\pi}{4}\right) - \left(0\right)
Fundamental theorem of calculus: plug in the limits.
= \frac{\pi}{4} \approx 0.78540
Simplify.

Reveal the answer

\frac{\pi}{4}