Evaluate the following integrals: ^7x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\cos^7\text{x}\text{ dx}$

✓

Answer

$\int\cos^7\text{x}\text{ dx}$
$=\int\cos^6\text{x}\cdot\cos\text{x}\text{ dx}$
$=\int(\cos^2\text{x})^3\cos\text{x}\text{ dx}$
$=\int(1-\sin^2\text{x})^3\cdot\cos\text{x}\text{ dx}$
Let $\sin\text{x}=\text{t}$
$\cos\text{x}\text{ dx}=\text{dt}$
Now, $\int(1-\sin^2\text{x})^3\cdot\cos\text{x}\text{ dx}$
$=\int(1-\text{t}^2)^3\text{dt}$
$=\int\big(1-\text{t}^6-3\text{t}^2+3\text{t}^4)\text{dt}$
$=\Big[\text{t}-\frac{\text{t}^7}{\text{7}}-\frac{3\text{t}^3}{3}+\frac{3\text{t}^5}{5}\Big]+\text{C}$
$=\sin\text{x}-\frac{1}{7}\sin^7\text{x}-\sin^3\text{x}+\frac{3}{5}\sin^5\text{x}+\text{C}$

Need a full question paper?

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

Start Generating Free

Explore more

More "3 Marks Question" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A function f(x) is defined as,
$\text{f}\text{(x)}=\begin{cases}\frac{\text{x}^2-\text{x}-6}{\text{x}-3}&; &\text{if} \text{x}\neq3\\5 &;&\text{if}\text{ x}=3\end{cases}$
show that f(x) is continuous that x = 3.

Let $\text{A}=\begin{bmatrix}2&-3\\-7&5\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&0\\2&-4\end{bmatrix},$ verify that
$(\text{A}+\text{B})^\text{T}=\text{A}^\text{T}+\text{B}^\text{T}$

Using differentials, find the approximate value of each of the following up to 3 places of decimal:
$(0.999)^{\frac{1}{10}}$

If $\overrightarrow{\text{PQ}}=3\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}}$ and the coordinates of P are (1, -1, 2), find the coordinates of Q.

A piece of ice is in the form of a cube melts so that the percentage error in the edge of cube is a, then find the percentage error in its volume.

Evaluate the following integrals:
$\int \frac{\text{1}}{\sqrt{\text{x}} + \text{x}} \text{ dx}$

Evaluate: $\int \frac{1}{\cos x-\sin x} d x$

Find the mean and standard deviation of the following probability distributions:

$x_i$	-1	0	1	2	3
$p_i$	0.3	0.1	0.1	0.3	0.2

Find the matrix A such that
$ \begin{bmatrix}4\\1\\3\end{bmatrix}\text{A}=\begin{bmatrix}-4&8&4\\-1&2&1\\-3&6&3\end{bmatrix}$

A doctor claims that $60 \%$ of the patients he examines are allergic to some type of weed. What is the probability that exactly $3$ of his next $4$ patients are allergic to weeds?