Solve ∫ (from 0 to 5) of 3sec^6d^2 wrt x

$$\int _ { 0 } ^ { 5 } 3 \sec ^ { 6 } d ^ { 2 } d x$$

Evaluate

$$\frac{15}{\left(\cos(d^{2})\right)^{6}}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3\times \left(\frac{1}{\cos(d^{2})}\right)^{6}\mathrm{d}x$$

Find the integral of \frac{3}{\left(\cos(d^{2})\right)^{6}} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{3}{\left(\cos(d^{2})\right)^{6}}x$$

Simplify.

$$\frac{3x}{\left(\cos(d^{2})\right)^{6}}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$3\times 5\left(\cos(d^{2})\right)^{-6}+0\times 3\left(\cos(d^{2})\right)^{-6}$$

Simplify.

$$\frac{15}{\left(\cos(d^{2})\right)^{6}}$$

Differentiate w.r.t. d

$$\frac{180d\tan(d^{2})}{\left(\cos(d^{2})\right)^{6}}$$