Question

Integrate 1/(1+x2) for limit (0,1).​

Answer 1

SOLUTION

TO INTEGRATE

$\displaystyle \sf{ \int\limits_{0}^{1} \frac{1}{1 + {x}^{2} } \, dx }$

EVALUATION

PROCESS : 1

$\displaystyle \sf{ \int\limits_{0}^{1} \frac{1}{1 + {x}^{2} } \, dx }$

$= \displaystyle \sf{{ \tan}^{ - 1} x \: \bigg|_0^1 }$

$= \displaystyle \sf{ { \tan}^{ - 1}(1) - { \tan}^{ - 1} (0) \: }$

$= \displaystyle \sf{ \frac{\pi}{4} - 0}$

$= \displaystyle \sf{ \frac{\pi}{4}}$

PROCESS : 2 ( Using Indefinite Integral )

$\displaystyle \sf{ \int\frac{1}{1 + {x}^{2} } \, dx }$

$\sf{ = { \tan}^{ - 1}x + c \: }$

Where C is integration constant

Hence

$\displaystyle \sf{ \int\limits_{0}^{1} \frac{1}{1 + {x}^{2} } \, dx }$

$= \displaystyle \sf{{ \tan}^{ - 1} x + c \: \bigg|_0^1 }$

$= \displaystyle \sf{ { \tan}^{ - 1}(1) - { \tan}^{ - 1} (0) + c - c\: }$

$= \displaystyle \sf{ \frac{\pi}{4} - 0 + 0}$

$= \displaystyle \sf{ \frac{\pi}{4}}$

━━━━━━━━━━━━━━━━

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Answer 2

Answer:

o

Step-by-step explanation:

SOLUTION

TO INTEGRATE

\displaystyle \sf{ \int\limits_{0}^{1} \frac{1}{1 + {x}^{2} } \, dx }

0

∫

1

1+x

2

1

dx

EVALUATION

PROCESS : 1

\displaystyle \sf{ \int\limits_{0}^{1} \frac{1}{1 + {x}^{2} } \, dx }

0

∫

1

1+x

2

1

dx

= \displaystyle \sf{{ \tan}^{ - 1} x \: \bigg|_0^1 }=tan

−1

x

∣

∣

∣

∣

∣

0

1

= \displaystyle \sf{ { \tan}^{ - 1}(1) - { \tan}^{ - 1} (0) \: }=tan

−1

(1)−tan

−1

(0)

= \displaystyle \sf{ \frac{\pi}{4} - 0}=

4

π

−0

= \displaystyle \sf{ \frac{\pi}{4}}=

4

π

PROCESS : 2 ( Using Indefinite Integral )

\displaystyle \sf{ \int\frac{1}{1 + {x}^{2} } \, dx }∫

1+x

2

1

dx

\sf{ = { \tan}^{ - 1}x + c \: }=tan

−1

x+c

Where C is integration constant

Hence

\displaystyle \sf{ \int\limits_{0}^{1} \frac{1}{1 + {x}^{2} } \, dx }

0

∫

1

1+x

2

1

dx

= \displaystyle \sf{{ \tan}^{ - 1} x + c \: \bigg|_0^1 }=tan

−1

x+c

∣

∣

∣

∣

∣

0

1

= \displaystyle \sf{ { \tan}^{ - 1}(1) - { \tan}^{ - 1} (0) + c - c\: }=tan

−1

(1)−tan

−1

(0)+c−c

= \displaystyle \sf{ \frac{\pi}{4} - 0 + 0}=

4

π

−0+0

= \displaystyle \sf{ \frac{\pi}{4}}=

4

π

━━━━━━━━━━━━━━━━

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a

∫ x/2+x⁸ dx , Evaluate it.

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