The equation $10x^4-3x^2-1=0$ has

1. Basics of Definite Integrals:

The fundamental theorem of calculus:

If f is continuous on a,b, then ${\int }_a^{b}f\left(x\right)dx=F\left(b\right)-F\left(a\right)$, where $F^{\text{'}}=f.$

Note:

If ${\int }_a^{b}f\left(x\right)dx=0 \Rightarrow f\left(x\right)=0$ has at least one root lying on $\left(a,b\right)$, provided f is a continuous function on a,b.

2. Properties of Definite Integral:

(i) ${\int }_a^{b}f\left(x\right)dx ={\int }_a^{b}f\left(t\right)dt$.

(ii) ${\int }_a^{b}f\left(x\right)dx=-{\int }_b^{a}f\left(x\right)dx$.

(iii) ∫abfxdx=∫acfxdx+∫cbfxdx.

(iv) ${\int }_{-a}^{a}f\left(x\right)dx={\int }_0^a\left[f\left(x\right)+f\left(-x\right)\right]dx=0;$ if fx is an odd function, i.e., f-x=-fx.

(v) ${\int }_{-a}^{a}f\left(x\right)dx={\int }_0^a\left[f\left(x\right)+f\left(-x\right)\right]dx=2{\int }_0^{a}f\left(x\right)dx;$if fx is an even function, i.e., f-x=fx.

(vi) ${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f\left(a+b-x\right)dx,$ in particular ∫0afxdx=∫0afa-xdx.

(vii) ∫02afxdx=∫0afxdx+∫0af2a-xdx=2∫0afxdx;if f2a-x=fx

(viii) ∫02afxdx=∫0afxdx+∫0af2a-xdx=0;if f2a-x=-fx.

(ix) ∫0nTfxdx=n∫0Tfxdx, n∈Z; where $T$ is the period of the function, i.e., fT+x=fx.

(x) ∫a+nTb+nTfxdx=∫abfxdx, where fx is periodic with period T & n∈Z.

(xi) ∫mTnTfxdx=n-m∫0Tfxdx,n,m∈Z, if fx is periodic with period T.

3. Definite Integral as Limit of a Sum:

${\int }_a^{b}f\left(x\right)dx=\underset{n\to \infty }{\lim }h\left[f\left(a\right)+f\left(a+h\right)+f\left(a+2h\right)+\dots +f\left(a+\underset{\_}{(n-1)h}\right)\right]=$$\mathrm{ }{\underset{h\to 0}{lim}}_{}h{\Sigma }_{r=0}^{n-1}f(a+rh), where b-a=nh$

If a=0 & b=1, then, $\underset{n\to \infty }{lim}h{\Sigma }_{r=0}^{n-1}f\left(rh\right)=\underset{n\to \infty }{lim}\left(\frac{1}{n}\right) {\Sigma }_{r=1}^{n-1}f\left(\frac{r}{n}\right)={\int }_0^{1}f\left(x\right)dx;$ where nh=1

4. Derivative of Antiderivative Function (Newton-Leibnitz’s Formula):

If $h\left(x\right) \&\hspace{0.17em}g\left(x\right)$ are differentiable functions of x, then $\frac{d}{dx}{\int }_{g\left(x\right)}^{h\left(x\right)}f\left(t\right)dt=f\left[h\left(x\right)\right]h^{\text{'}}\left(x\right)-f\left[g\left(x\right)\right]·g^{\text{'}}\left(x\right)$.

5. Reduction and Walli’s Formula:

(i) ${\int }_0^{\frac{\pi }{2}}{\sin }^{n}xdx={\int }_0^{\frac{\pi }{2}}{\cos }^{n}xdx=\frac{\left(n-1\right)\left(n-3\right)\dots \left(1 or 2\right)}{n\left(n-2\right)\dots \dots \left(1 or 2\right)}K,$

where K=π2, if nis even;

and K=1 ,if nis odd.

(ii) ${\int }_0^{\frac{\pi }{2}}{\sin }^{n}x·co{s}^{m}xdx=\frac{\left[\left(n-1\right)\left(n-3\right)\left(n-5\right)\dots .1 or 2\right]\left[\left(m-1\right)\left(m-3\right)\dots .1 or 2\right]}{\left(m+n\right)\left(m+n-2\right)\left(m+n-4\right)\dots 1 or 2}K$,

where K=π2, if both m and n are even;

and $K=1,$ otherwise.

6. Estimation of Definite Integral:

(i) If $f\left(x\right)$ is continuous on a,b and its range on this interval is m,M, then mb-a≤∫abfxdx≤Mb-a.

(ii) If fx≤ϕx for $a\le x\le b,$ then ∫abfxdx≤∫abϕxdx.

(iii) ∫abfxdx≤∫abfxdx.

(iv) If fx≥0 on the interval a,b, then ∫abfxdx≥0.

(v) fx and gx are two continuous function on a,b, then ∫abfxgxdx≤∫abf2xdx∫abg2xdx.

(vi) Trapezoidal approximation:

Let fx be a continuous function on the interval a, b. Dividing the interval a, b into n equal subintervals, each of width $\Delta x=\frac{\left(b-a\right)}{n}$ such that a=x0<x1<x2<x3…xn=b, then ∫abfxdx ≈I = Δxfx0+2fx1+2fx2+......+2fxn-1+xn2.

(vii) Simpson’s approximation rule:

Let fx be a continuous function on a, b,then ${\int }_a^{b}f\left(x\right)dx\approx \frac{\left(b-a\right)}{6}\left(f\left(a\right)+4f\left(\frac{b+a}{2}\right)+f\left(b\right)\right)$.

7. Length of Arc:

Length of the arc of $y=f\left(x\right)$ from x=a to x=b is ∫ab1+dydx2dx