Evaluate - -22x2-1-5 x d x

∫^2_ 2x^2/1+5^xdx ......(i) [∴∫^b_a f(x)dx = ∫^b_af(a+b x)dx] ⇒ I = ∫^2_ 2( 2+2 x)^2/1+5^ 2+2 xdx = ∫^2_ 2x^2/1+5^ xdx....(ii) Adding (i) and (ii), 2I = ∫^2 ...

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Standard XII

Mathematics

Functions

Evaluate : ∫-...

Question

Evaluate : $\int_{- 2}^{2} \frac{x^{2}}{1 + 5^{x}} d x .$

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Solution

$\int_{- 2}^{2} \frac{x^{2}}{1 + 5^{x}} d x . . . . . . (i) [∴ \int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x]$

$\Rightarrow I = \int_{- 2}^{2} \frac{(- 2 + 2 - x)^{2}}{1 + 5^{- 2 + 2 - x}} d x = \int_{- 2}^{2} \frac{x^{2}}{1 + 5^{- x}} d x . . . . (i i)$

Adding (i) and (ii), $2 I = \int_{- 2}^{2} x^{2} d x = \frac{1}{3} [x^{3}]_{- 2}^{2} = \frac{1}{3} [8 + 8] ∴ I = \frac{8}{3} .$

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Evaluate : ∫2−2x21+5xdx.

∫2−2x21+5xdx......(i) [∴∫baf(x)dx=∫baf(a+b−x)dx] ⇒I=∫2−2(−2+2−x)21+5−2+2−xdx=∫2−2x21+5−xdx....(ii) Adding (i) and (ii), 2I=∫2−2x2dx=13[x3]2−2=13[8+8] ∴I=83.

Evaluate : $\int_{- 2}^{2} \frac{x^{2}}{1 + 5^{x}} d x .$