Evaluate _ -1 ^ 1 5 x^ 4 x^ 5 +1 d x - Vidyadip

Question

Evaluate $\int_{-1}^{1} 5 x^{4} \sqrt{x^{5}+1} d x$

✓

Answer

Put t = x⁵ + 1, then dt = 5x⁴ dx.
Therefore, $\int 5 x^{4} \sqrt{x^{5}+1} d x$ = $\int \sqrt{t} d t=\frac{2}{3} t^{\frac{3}{2}}=\frac{2}{3}\left(x^{5}+1\right)^{\frac{3}{2}}$
Hence, $\int_{-1}^{1} 5 x^{4} \sqrt{x^{5}+1} d x$ = $\frac{2}{3}\left[\left(x^{5}+1\right)^{\frac{3}{2}}\right]_{-1}^{1}$
= $\frac{2}{3}\left[\left(1^{5}+1\right)^{\frac{3}{2}}-\left((-1)^{5}+1\right)^{\frac{3}{2}}\right]$
= $\frac{2}{3}\left[2^{\frac{3}{2}}-0^{\frac{3}{2}}\right]=\frac{2}{3}(2 \sqrt{2})=\frac{4 \sqrt{2}}{3}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "1 Marks" from Integrals→Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Examine the function for continuity.
$f(x)=\frac{1}{x-5}$, x $\neq$ 5

Find the absolute maximum and minimum values of a function f given by
f (x) = 2x³ – 15x² + 36x +1 on the interval [1, 5].

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\text{x}^2\Big(\frac{\text{d}^2\text{y}}{\text{dx}}\Big)^3+\text{y}\Big(\frac{\text{dy}}{\text{dx}}\Big)^4+\text{y}^4=0$

Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find:

Which elements of N are invertible for the operation *?

Find the direction cosines of normals of the plane $\vec{r} \cdot(3 \hat{i}+\hat{j}+4 \hat{k})=5$.

Evaluate the following:
$\text{cosec}^{-1}\Big\{\text{cosec}\Big(-\frac{9\pi}{4}\Big)\Big\}$

Find the rate of change of the area of a circle with respect to its radius r when r = 4 cm.

If $A =\left[\begin{array}{cc}3 & -3 \\ 3 & 3\end{array}\right], B =\left[\begin{array}{ll}6 & 3 \\ 4 & 2\end{array}\right]$ and $C =\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$ then which of the matrix will be invertible matrix?

If A is an invertible matrix of order 2, then det (A^–1) is equal to

Evaluate $\int _ { 0 } ^ { 1 } \frac { \tan ^ { - 1 } x } { 1 + x ^ { 2 } } dx$