Value of -101 - x - x2 - x3 1 - 3x - 5x2 - 7x3dx is

We have, ∫^1_0(1 + x + x^2 + x^3) (1 + 3x + 5x^2 + 7x^3)dx Let x = t^2 ⇒ dx = 2t dt I = 2 ∫^1_0(t + t^3 + t^5 + t^7)(1 + 3t^2 + 5t^4 + 7t^6)dt Let (t + t^3 + t^ ...

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Byju's Answer

Standard XIII

Mathematics

Integration by Substitution

Value of ∫101...

Question

Value of $\int_{0}^{1} (1 + x + x^{2} + x^{3}) (1 + 3 x + 5 x^{2} + 7 x^{3}) d x$ is

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Solution

We have,
$\int_{0}^{1} (1 + x + x^{2} + x^{3}) (1 + 3 x + 5 x^{2} + 7 x^{3}) d x$
$Let x = t^{2} \Rightarrow d x = 2 t d t$
$I = 2 \int_{0}^{1} (t + t^{3} + t^{5} + t^{7}) (1 + 3 t^{2} + 5 t^{4} + 7 t^{6}) d t$
$Let (t + t^{3} + t^{5} + t^{7}) = u \Rightarrow (1 + 3 t^{2} + 5 t^{4} + 7 t^{6}) d t = d u$
$= 2 \int_{0}^{4} u . d u = (u^{2})_{0}^{4} = 16$

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Similar questions

Q. Value of

\int_{0}^{1} (1 + x + x^{2} + x^{3}) (1 + 3 x + 5 x^{2} + 7 x^{3}) d x

The factors of x⁴ + x² + 25 are

(a) (x² + 3x + 5) (x² − 3x + 5)

(b) (x²+ 3x + 5) (x² + 3x − 5)

(c) (x² + x +5) (x² − x + 5)

(d) none of these

Q. If the matrix

[\begin{matrix} 5 x & 2 \\ - 10 & 1 \end{matrix}]

is singular, find the value of x.

Q. State True or False.

(x^{2} - 3 x)^{2} - 5 (x^{2} + 3 x) - y (x^{2} + 3 x) + 5 y

= (x^{2} + 3 x - 5) (x^{2} + 3 x - y)

Q. If

a x^{3} - 5 x^{2} + x + p

is divisible by

x^{2} - 3 x + 2

then find the values of

a

and

p

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Value of ∫10(1+x+x2+x3)(1+3x+5x2+7x3)dx is

We have, ∫10(1+x+x2+x3)(1+3x+5x2+7x3)dx Let x=t2⇒dx=2t dt I=2∫10(t+t3+t5+t7)(1+3t2+5t4+7t6)dt Let (t+t3+t5+t7)=u⇒(1+3t2+5t4+7t6) dt=du =2∫40u.du=(u2)40=16

Value of $\int_{0}^{1} (1 + x + x^{2} + x^{3}) (1 + 3 x + 5 x^{2} + 7 x^{3}) d x$ is