Question

Try to solve this integral,
$\displaystyle \tiny{ \int\left(\sqrt{1-x^2} + \sqrt{x^2+1} + x^2 \ln x + \dfrac{\tan^{-1}x}{x^2} + x^\frac{\sqrt{x}}{\ln x} + \dfrac{x^2 + 1}{x^3 - 2x} + \sin^4x \cos^2x + \dfrac{1}{2 + \cos x} + \tan^{10} x + x^x\right) dx}$

Correct answer would be appreciable and will be rewarded as brainliest.

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Answer 1

Step-by-step explanation:

Now, we can write is as ,

\begin{gathered} \longrightarrow \sf{\quad { \dfrac{30}{6} + \dfrac{12\sqrt{6} }{6} }} \\ \end{gathered}

⟶

6

30

+

6

12

6

Performing division.

\begin{gathered} \longrightarrow \quad \underline{\boxed{\sf { 5 + 2\sqrt{6} }}} \\ \end{gathered}

⟶

5+2

6

Now, according to the question,

\begin{gathered} \longrightarrow \sf{\quad { \dfrac{3\sqrt{2}-2\sqrt{3}}{3\sqrt{2}+2\sqrt{3}} + \dfrac{3\sqrt{2}+2\sqrt{3}}{3\sqrt{2}-2\sqrt{3}} }} \\ \end{gathered}

⟶

3

2

+2

3

3

2

−2

3

+

3

2

−2

3

3

2

+2

3

Substitute the rationalised form of these two terms.

\begin{gathered} \longrightarrow \sf{\quad {(5 -2\sqrt{6} ) + (5 + 2\sqrt{6} ) }} \\ \end{gathered}

⟶(5−2

6

)+(5+2

6

)

Removing the brackets.

\begin{gathered} \longrightarrow \sf{\quad {5 -2\sqrt{6} + 5 + 2\sqrt{6} }} \\ \end{gathered}

⟶5−2

6

Performing addition and subtraction.

\begin{gathered} \longrightarrow \quad\underline{\boxed { \pmb{\mathfrak{10}} }} \\ \end{gathered}

⟶

10

10

Therefore, the required answer