Evaluate the following integrals: ( )dx - Vidyadip

Question

Evaluate the following integrals:

$\int\sin\text{x}\log(\cos\text{x})\text{dx}$

✓

Answer

Let $\text{I}=\int\sin\text{x}\cdot\log(\cos\text{x})\text{dx}$
Let $\cos\text{x = t}$
$\Rightarrow-\sin\text{x dx =}\text{ dt}$
$\Rightarrow\sin\text{x dx =}-\text{dt}$
$\therefore\text{I}=-\int\log\text{t dt}$
$=-\int1\cdot\log\text{t dt}$
Taking log t as the first function and 1 as the second function.
$=\log\text{t}\int1\text{dt}-\int\big\{\frac{\text{d}}{\text{dt}}(\log\text{t})\int1\text{dt}\big\}\text{dt}$
$=-[\log\text{t}\cdot\text{t}-\int\frac{1}{\text{t}}\times\text{t dt}]$
$=-[\log\text{t}\cdot\text{t}-\text{t}]+\text{C}$
$=-\text{t}(\log\text{t}-1)+\text{C} \dots(1)$
Substituting the value of t in eq (1)
$=-\cos\text{x}\{\log(\cos\text{x})-1\}+\text{C}$
$=\cos\text{x}\{1-\log(\cos\text{x})\}+\text{C}$

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 "4 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

using interation, find the area of the region bounded by the triangle ABC vertices A, B, C are (-1, 1), (0, 5) and (3, 2) respectivrly.

Evaluate the following integrals:
$\int\frac{\text{x}^2}{\text{x}^4+\text{x}^2-2}\ \text{dx}$

Find the equation of the plane passing through the line of intersection of the planes 2x - 7y + 4z = 0, 3x - 5y + 4z + 11 = 0 and the point (-2, 1, 3).

Show that the relation R on the set A = {x ∈ Z; 0 ≤ x ≤ 12}, given by R = {(a, b): a = b}, is an equivalence relation. Find the set of all elements related to 1.

Find the distance of the point (3, 3, 3) from the plane $\vec{\text{r}}\cdot(5\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}})+9=0$

If $\sqrt{1-\text{x}^2}+\sqrt{1-\text{y}^2}=\text{a}(\text{x}-\text{y}),$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sqrt{1-\text{y}^2}}{1-\text{x}^2}$

Find the coordinate of the point P where the line through $\text{A(3, – 4, –5) and B (2, –3, 1)}$ crosses the plane passing through three points L(2, $\text{2, 1), M(3, 0, 1) and N(4, –1, 0).}$ Also, find the ratio in which P divides the line segment AB.

Using differentials, find the approximate values of the following:
$(255)^{\frac{1}{4}}$

If A and B are two events such that $2\text{P(A)}=\text{P(B)}=\frac{5}{13}$ and $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{2}{5}$ find $\text{P}(\text{A}\cap\text{B}).$

Show that the differential equation $2 x y+y^{2}-2 x^{2} \frac{d y}{d x}=0$, is homogenous and find the particular solution, given that y = 2 when x = 1.