Question

evaluate the following integers ​

Answer 1

REFER TO THE ATTACHMENT FOR THE ANSWER

Answer 2

Solution:

18)

$\int \: {tan}^{ - 1}( \sqrt{x} )dx$

Here , we use the formula of integration by parts :

$\boxed{\int uv \: dx = u \int \: v \: dx - \int \: ( \frac{du}{dx} \int \: v \: dx) \: dx }$

Here , we let the other function as unity which taken as 2nd function :

$\int \: {tan}^{ - 1} ( \sqrt{x} ).(1)dx \\ \\ \implies {tan}^{ - 1} ( \sqrt{x} ) \int \: 1 \: dx - \int( \frac{d}{dx} ( {tan}^{ -1 } ( \sqrt{x} ) \int \: 1 \: dx)dx \\ \\ \implies \: {tan}^{ - 1} ( \sqrt{x} )(x) - \int \: ( \frac{1}{1 + {( \sqrt{x} )}^{2} } (x))dx \ \\ \\ \implies \: x \: {tan}^{ - 1} ( \sqrt{x} ) - \int \: \frac{x}{1 + x} dx \\ \\ \implies \: \: x \: {tan}^{ - 1} (\sqrt{x} ) - \int\: (1 - \frac{1}{x + 1} )dx \\ \\ \implies \: x \: {tan}^{ - 1} ( \sqrt{x} ) - x + log(x + 1) + c$

19) Solution refer to the attachment.