Question

How many of the following statements are correct?

1. $\int x^{2}dx=x^3$

2. $\int \text{sin}xdx=\text{cos}x$

3. $\int e^{x}dx=\ln x$

4. $\int \frac{1}{x}dx=x^0$

5. $\int \text{tanx}dx={\text{sec}}^{2}x$

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Answer 1

We saw that integration is inverse process of differentiation. So to check if the following statements are correct, we will differentiate the function on the RHS, and check if we get the function on the LHS in integral.

1. Derivative of $x^3=3x^2$ So the first relation is not correct.
2. Derivative of $\cos x=-\sin x\Rightarrow$ in correct
3. Derivative of $\ln x=\frac{1}{x}\Rightarrow$ incorrect
4. Derivative of $x^0$= Derivative of $1=0\Rightarrow$ Incorrect
5. Derivative of ${\sec }^{2}x=2\sec x\sec x\tan x\Rightarrow$ Incorrect
   So, all the integrals given are wrong.

=> answer = 0

It’s also important to know the integral of all these functions for solving other questions.

Correct relation is as follows

1. $\int x^2=x^3/3+C$
2. $\int \sin x=-cosx+C$
3. $\int e^x=e^x+C$
4. $\int \frac{1}{x}=\ln |x|+C$
5. $\int \tan x=\ln |\sec x|+C$

You have seen all the integrals except the last one, which we will see in standard formulae - 2