List of practice Questions

Integrate the function: \(\frac{2x}{1+x^2}\)

Find the following integral: \(\int \frac{2-3 \sin x}{\cos^2 x}dx\)

Find the following integral: \(\int \frac{\sec^2 x}{\cosec^2 x}dx\)

Find the following integral: \(\int \sec x(\sec x +\tan x)dx\)

Find the following integral: \(\int (2x^2-3sinx+5\sqrt x)dx\)

Find the following integral: \(\int (2x-3\cos x +e^x)dx\)

Find the following integral: \(\int \sqrt x(3x^2+2x+3)dx\)

Find the following integral: \(\int (1-x)\sqrt xdx\)

Find the following integral: \(\int \frac{x^3-x^2+x-1}{x-1}dx\)

Find the following integral: \(\int \frac{x^3+3x+4}{\sqrt x}dx\)

Find the following integral: \(\int \frac{x^3+5x^2-4}{x^2}dx\)

Find the following integral: \(\int \bigg( \sqrt x - \frac{1}{\sqrt x}\bigg)^2dx\)

Find the following integral: \(\int x^2\bigg(1-\frac{1}{x}\bigg)dx\)

Find the following integral: \(\int(4e^{3x}+1)dx\)

Differentiate w.r.t. x the function:\(x^{x^2-3}+(x-3)^{x^2},for\, x>3\)

Differentiate w.r.t. x the function:\((sinx-cosx)^{(sinx-cosx)},\frac{π}{4}<x<\frac{3π}{4}\)

Differentiate w.r.t. x the function:\(cos(a\,cosx+b\,sinx)\),for some constant \(a\) and \(b\). 

Differentiate w.r.t. \(x\) the function: \((logx)^{logx},x>1\)

Differentiate w.r.t. \(x\) the function: \(cot^{-1}\bigg[\frac{\sqrt{1+sinx}+\sqrt{1-sinx}}{\sqrt{1+sinx}-\sqrt{1-sinx}}\bigg],0<x<\frac{x}{2}\)

Differentiate w.r.t. \(x\) the function: \(\frac{cos^{-1}\frac{x}{2}}{\sqrt{2x+7}},-2<x<2\)

Differentiate w.r.t. \(x\) the function: \(sin^{-1}(x\sqrt{x}),0≤x≤1\)

Differentiate w.r.t. \(x\) the function: \((5x)^{3cos2x}\)

Differentiate w.r.t. \(x\) the function: \(sin^3x+cos^6x\)

Prove that sin 2x+2sin 4x+sin6x=4cos² x sin4x.

Write the first five terms of each of the sequences in Exercises 11 to 13 and obtain the corresponding series: a1 = a2 = 2, an = an-1 -1, n > 2.