Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\int x^3 \cdot \sin \left( x^4 + 1 \right) dx\]
\[\text{Let x}^4 + 1 = t\]
\[ \Rightarrow 4 x^3 = \frac{dt}{dx}\]
\[ \Rightarrow x^3 \text{dx} = \frac{dt}{4}\]
\[Now, \int x^3 \cdot \sin \left( x^4 + 1 \right) dx\]
\[ = \frac{1}{4}\int\sin \left( t \right) dt\]
\[ = \frac{1}{4}\left[ - \cos t \right] + C\]
\[ = \frac{1}{4}\left[ - \cos \left( x^4 + 1 \right) \right] + C\]
APPEARS IN
संबंधित प्रश्न
Evaluate `int_1^3(e^(2-3x)+x^2+1)dx` as a limit of sum.
Evaluate the following definite integrals as limit of sums.
`int_a^b x dx`
Evaluate the following definite integrals as limit of sums.
`int_0^5 (x+1) dx`
Evaluate the following definite integrals as limit of sums.
`int_2^3 x^2 dx`
Evaluate the following definite integrals as limit of sums `int_(-1)^1 e^x dx`
Evaluate the definite integral:
`int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx`
Evaluate the definite integral:
`int_0^(pi/2) (cos^2 x dx)/(cos^2 x + 4 sin^2 x)`
Prove the following:
`int_0^(pi/4) 2 tan^3 xdx = 1 - log 2`
Prove the following:
`int_0^1sin^(-1) xdx = pi/2 - 1`
`int dx/(e^x + e^(-x))` is equal to ______.
`int (cos 2x)/(sin x + cos x)^2dx` is equal to ______.
Evaluate the following integral:
Evaluate the following integrals as limit of sums:
Evaluate `int_1^4 ( 1+ x +e^(2x)) dx` as limit of sums.
Solve: (x2 – yx2) dy + (y2 + xy2) dx = 0
Evaluate the following as limit of sum:
`int _0^2 (x^2 + 3) "d"x`
Evaluate the following as limit of sum:
`int_0^2 "e"^x "d"x`
Evaluate the following:
`int_0^2 ("d"x)/("e"^x + "e"^-x)`
Evaluate the following:
`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`
Evaluate the following:
`int_0^1 (x"d"x)/sqrt(1 + x^2)`
Evaluate the following:
`int_0^pi x sin x cos^2x "d"x`
The value of `int_(-pi)^pi sin^3x cos^2x "d"x` is ______.
The limit of the function defined by `f(x) = {{:(|x|/x",", if x ≠ 0),(0",", "otherwisw"):}`
Let f: (0,2)→R be defined as f(x) = `log_2(1 + tan((πx)/4))`. Then, `lim_(n→∞) 2/n(f(1/n) + f(2/n) + ... + f(1))` is equal to ______.
The value of `lim_(n→∞)1/n sum_(r = 0)^(2n-1) n^2/(n^2 + 4r^2)` is ______.
`lim_(n→∞){(1 + 1/n^2)^(2/n^2)(1 + 2^2/n^2)^(4/n^2)(1 + 3^2/n^2)^(6/n^2) ...(1 + n^2/n^2)^((2n)/n^2)}` is equal to ______.
