Evalute the following integrals: ( + ) dx - Vidyadip

Question

Evalute the following integrals:
$\int\frac{\sec\text{x}}{\log(\sec\text{x}+\tan\text{x})}\text{dx}$

✓

Answer

Note: Here, we are considering $\log\text{x}$ as $\log_\text{e}\text{x}$
Let $\text{I}=\int\frac{\sec\text{x}}{\log(\sec\text{x}+\tan\text{x})}\text{dx}$
Putting $\log(\sec\text{x}+\tan\text{x})=\text{t}$
$\Rightarrow\frac{\sec\text{x}\tan\text{x}+\sec^2\text{x}}{\sec\text{x}+\tan\text{x}}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\sec\text{x}\frac{(\sec\text{x}+\tan\text{x})}{\sec\text{x}+\tan\text{x}}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\sec\text{x }\text{dx}=\text{dt}$
$\therefore\text{I}=\int\frac{\text{dt}}{\text{t}}$
$=\log|\text{t}|+\text{C}$
$=\log|\log(\sec\text{x}+\tan\text{x})|+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following intregals:
$\int\frac{\cos\text{x}}{(1-\sin\text{x})^3(2+\sin\text{x})}\ \text{dx}$

Using differentials, find the approximate values of the following:
$\log_\text{e}10.02$ it being given that $\log_\text{e}10=2.3026$

Find the equation of the line passing through the points $(1, -1, 1)$ and perpendicular to the lines joining the points $(4, 3, 2), (1, -1, 0)$ and $(1, 2, -1) (2, 1, 1).$

Let f = {(1, -1), (4, -2), (9, -3), (16, 4)} and g = {(-1, -2), (-2, -4), (-3, -6), (4, 8)}. Show that gof is defined while fog is not defined. Also, find gof.

If either $\vec{\text{a}}=\vec{0}$ or $\vec{\text{b}}=\vec{0},$ then $\vec{\text{a}}.\vec{\text{b}}=0.$ But the converse need not be true. Justify your answer with an example.

Solve the following initial value problems:
$\text{x}(\text{x}^2+3\text{y}^2)\text{dx}+\text{y}(\text{y}^2+3\text{x}^2)\text{dy}=0,\text{y}(1)=1$

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
$\text{f}(\text{x})=\text{x}+\sqrt{1-\text{x}},\text{x}\leq 1$

Find the feasible solution of the following inequations graphically.3x + 4y ≥ 12, 4x + 7y ≤ 28, y ≥ 1, x ≥ 0

Compute the adjoint of the following matrices:$\begin{bmatrix} 1 & 2 & 5 \\ 2 & 3 & 1 \\ -1 & 1 & 1 \end{bmatrix}$
Verify that (adjoint A)A = |A|I = A (adjoint A) for the above matrices.

Find the equation of the tangent line to the curve $y = x^2 - 2x + 7$ which is perpendicular to the line $5y - 15x = 13.$