Advertisements
Advertisements
Question
Evaluate the following using properties of definite integral:
`int_(- pi/4)^(pi/4) x^3 cos^3 x "d"x`
Sum
Advertisements
Solution
Let f(x) = `x^3 cos^3x`
f(– x) = `(- x)^3 cos^3 (- x)`
= `- x^3 cos^3 x`
= `- "f"(x)`
Here f(– x) = – f(x)
∴ f(x) is an odd function
∴ `int_(- pi/4)^(pi/4) x^3 cos^3 x "d"x` = 0
shaalaa.com
Definite Integrals
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int_0^\pi e^{2x} \cdot \sin\left( \frac{\pi}{4} + x \right) dx\]
\[\int_0^\frac{1}{2} \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}dx\]
\[\int\limits_0^\pi \frac{1}{5 + 3 \cos x} dx\]
\[\int_{- \frac{\pi}{2}}^\pi \sin^{- 1} \left( \sin x \right)dx\]
\[\int\limits_0^\pi x \log \sin x\ dx\]
Evaluate the following integral:
\[\int_{- a}^a \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right)d\theta\]
Prove that:
\[\int_0^\pi xf\left( \sin x \right)dx = \frac{\pi}{2} \int_0^\pi f\left( \sin x \right)dx\]
The value of \[\int\limits_0^1 \tan^{- 1} \left( \frac{2x - 1}{1 + x - x^2} \right) dx,\] is
The value of \[\int\limits_{- \pi/2}^{\pi/2} \left( x^3 + x \cos x + \tan^5 x + 1 \right) dx, \] is
Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`
