Question
Write a value of $\int\text{x}^2\sin\text{x}^3\text{ dx}$
✓
Answer
Let $\text{I}=\int\text{x}^2\sin\text{x}^3\text{ dx}$
Let $\text{x}^3=\text{t}$
$=3\text{x}^2\text{dx}=\text{dt}$
$=\text{x}^2\text{dx}=\frac{1}{3}\text{dt}$
$\therefore\ \text{I}=\frac{1}{3}\int\sin\text{t dt}$
$=\frac{1}{3}(-\cos\text{t})+\text{C}$
Hence, $\text{I}=\frac{-1}{3}\cos\text{x}^3+\text{C}$
Let $\text{x}^3=\text{t}$
$=3\text{x}^2\text{dx}=\text{dt}$
$=\text{x}^2\text{dx}=\frac{1}{3}\text{dt}$
$\therefore\ \text{I}=\frac{1}{3}\int\sin\text{t dt}$
$=\frac{1}{3}(-\cos\text{t})+\text{C}$
Hence, $\text{I}=\frac{-1}{3}\cos\text{x}^3+\text{C}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If $\vec{\text{a}},\vec{\text{b}}$ are two vectors, then write the truth value of the following statement:$\big|\vec{\text{a}}\big|=\big|\vec{\text{b}}\big|\Rightarrow\vec{\text{a}}=\pm\vec{\text{b}}$
→Find the expected value, variance, and standard deviation of the random variable whose p.m.f’s are given below:
Differentiate $\tan ^{-1}\left(\frac{\cos x}{1+\sin x}\right)$ w.r.t. $\sec ^{-1} x$.
→If two of the vertices of the triangle are A(3, 1, 4) and B(-4, 5, -3) and the centroid of a triangle is G(-1, 2, 1), then find the coordinates of the third vertex C of the triangle.
Verify Rolle’s theorem for the following functions.
f(x) = sin x + cos x + 7, x ∈ [0, 2π]
Find the cartesian equation of the line which passes through the point (-2, 4, -5) and parallel to the line given by $\frac{\text{x}+3}{3}=\frac{\text{y}-4}{5}=\frac{\text{z}+8}{6}.$
→Let * be a binary operation on Q - {-1} defined by a * b = a + b + ab for all a, b ∈ Q - {-1}. Then,
Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element.
Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element.
If $\bar{a}=\hat{i}+2 \hat{j}+3, \bar{b}=3 \hat{i}+2 \hat{j}$ and $\bar{c}=2 \hat{i}+\hat{j}+3$ then verify that $\bar{a} \times(\bar{b} \times \bar{c})=(\bar{a} \cdot \bar{c})$
→$\bar{b}-(\bar{a} \cdot \bar{b}) \bar{c}$
Dot product of a vector with vectore $\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}},2\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}$ and $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ are respectively 4, 0 and 2. Find the vector.
→If $\text{A}=\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\text{ and A (adj A =)}\begin{bmatrix} \text{k} & 0 \\ 0 & \text{k} \end{bmatrix},$ then find the value of k.
→